Just noticed you're from Texas, ajstoner. That being the case, your original question seemed to pertain to whether or not the turn coordinator was a required item, as well as whether or not the turn coordinator had a problem.
Without actually being able to see the airplane in question, it's impossible to diagnose your turn coordinator problem. It could be the instrument, a circuit breaker, a loose cannon plug, or perhaps an additional power switch for the turn coordinator on the panel (some aircraft use them to prevent the turn coordinator from running when it's not needed).
The day VFR instrumentation and equipment requirement to which you referred is found in 91.205:
§ 91.205 Powered civil aircraft with standard category U.S. airworthiness certificates: Instrument and equipment requirements.
(a) General. Except as provided in paragraphs (c)(3) and (e) of this section, no person may operate a powered civil aircraft with a standard category U.S. airworthiness certificate in any operation described in paragraphs (b) through (f) of this section unless that aircraft contains the instruments and equipment specified in those paragraphs (or FAA-approved equivalents) for that type of operation, and those instruments and items of equipment are in operable condition.
(b) Visual-flight rules (day). For VFR flight during the day, the following instruments and equipment are required:
(1) Airspeed indicator.
(2) Altimeter.
(3) Magnetic direction indicator.
(4) Tachometer for each engine.
(5) Oil pressure gauge for each engine using pressure system.
(6) Temperature gauge for each liquid-cooled engine.
(7) Oil temperature gauge for each air-cooled engine.
(8) Manifold pressure gauge for each altitude engine.
(9) Fuel gauge indicating the quantity of fuel in each tank.
(10) Landing gear position indicator, if the aircraft has a retractable landing gear.
(11) For small civil airplanes certificated after March 11, 1996, in accordance with part 23 of this chapter, an approved aviation red or aviation white anticollision light system. In the event of failure of any light of the anticollision light system, operation of the aircraft may continue to a location where repairs or replacement can be made.
(12) If the aircraft is operated for hire over water and beyond power-off gliding distance from shore, approved flotation gear readily available to each occupant and, unless the aircraft is operating under part 121 of this subchapter, at least one pyrotechnic signaling device. As used in this section, “shore” means that area of the land adjacent to the water which is above the high water mark and excludes land areas which are intermittently under water.
(13) An approved safety belt with an approved metal-to-metal latching device for each occupant 2 years of age or older.
(14) For small civil airplanes manufactured after July 18, 1978, an approved shoulder harness for each front seat. The shoulder harness must be designed to protect the occupant from serious head injury when the occupant experiences the ultimate inertia forces specified in §23.561(b)(2) of this chapter. Each shoulder harness installed at a flight crewmember station must permit the crewmember, when seated and with the safety belt and shoulder harness fastened, to perform all functions necessary for flight operations. For purposes of this paragraph—
(i) The date of manufacture of an airplane is the date the inspection acceptance records reflect that the airplane is complete and meets the FAA-approved type design data; and
(ii) A front seat is a seat located at a flight crewmember station or any seat located alongside such a seat.
(15) An emergency locator transmitter, if required by §91.207.
(16) For normal, utility, and acrobatic category airplanes with a seating configuration, excluding pilot seats, of 9 or less, manufactured after December 12, 1986, a shoulder harness for—
(i) Each front seat that meets the requirements of §23.785 (g) and (h) of this chapter in effect on December 12, 1985;
(ii) Each additional seat that meets the requirements of §23.785(g) of this chapter in effect on December 12, 1985.
(17) For rotorcraft manufactured after September 16, 1992, a shoulder harness for each seat that meets the requirements of §27.2 or §29.2 of this chapter in effect on September 16, 1991.
While the turn coordinator isn't required for day VFR, it's still got to work if it's installed, or be properly altered. Like most reguations, reference to one reg usually won't answer your question. The answer, then, is found in 91.213, which deals with inoperative equipment and instruments.
§ 91.213 Inoperative instruments and equipment.
(a) Except as provided in paragraph (d) of this section, no person may take off an aircraft with inoperative instruments or equipment installed unless the following conditions are met:
The point of that paragraph is that everything must work. Three types of relief are found. One is the MEL, if you have a minimum equipment list for your aircraft. It's got to be specifically designated for your airplane by registration and serial number, and approved by the FAA.
The second choice is removal or deactivation. This is the more common method in light, private airplanes. The third option is to obtain a special flight permit, sometimes called a "ferry permit."
§ 91.213 Inoperative instruments and equipment.
(d) Except for operations conducted in accordance with paragraph (a) or (c) of this section, a person may takeoff an aircraft in operations conducted under this part with inoperative instruments and equipment without an approved Minimum Equipment List provided—
(1) The flight operation is conducted in a—
(i) Rotorcraft, non-turbine-powered airplane, glider, lighter-than-air aircraft, powered parachute, or weight-shift-control aircraft, for which a master minimum equipment list has not been developed; or
(ii) Small rotorcraft, nonturbine-powered small airplane, glider, or lighter-than-air aircraft for which a Master Minimum Equipment List has been developed; and
(2) The inoperative instruments and equipment are not—
(i) Part of the VFR-day type certification instruments and equipment prescribed in the applicable airworthiness regulations under which the aircraft was type certificated;
(ii) Indicated as required on the aircraft's equipment list, or on the Kinds of Operations Equipment List for the kind of flight operation being conducted;
(iii) Required by §91.205 or any other rule of this part for the specific kind of flight operation being conducted; or
(iv) Required to be operational by an airworthiness directive; and
(3) The inoperative instruments and equipment are—
(i) Removed from the aircraft, the cockpit control placarded, and the maintenance recorded in accordance with §43.9 of this chapter; or
(ii) Deactivated and placarded “Inoperative.” If deactivation of the inoperative instrument or equipment involves maintenance, it must be accomplished and recorded in accordance with part 43 of this chapter; and
(4) A determination is made by a pilot, who is certificated and appropriately rated under part 61 of this chapter, or by a person, who is certificated and appropriately rated to perform maintenance on the aircraft, that the inoperative instrument or equipment does not constitute a hazard to the aircraft.
An aircraft with inoperative instruments or equipment as provided in paragraph (d) of this section is considered to be in a properly altered condition acceptable to the Administrator.
Note that to get the aircraft is in a "properly altered condition," one also needs to visit Part 43 to understand the maintenance and paperwork requirements...Part 91 only tells part of the story.