Right here's my long-winded tuppence around the topic.
1. Paper stick-ons used to be the norm for CDs. They aren't for DVDs. You may come across comments such as 'the labels put the DVDs out of balance'. I have no idea if that is true, however as everyone else has said just ignore stickies.
2. DIY.
Duplication of the disks. For a run of 100 it won't be economical for you to go to a professional duplicator for a glass master and duplication from that. (Usually this kicks in at runs of 500+.)
So you will have to do the duplication yourself on DVD+R or DVD-R disks that have a Printable Surface. Verbatim are the usual brand on the high street but there are other choices by mail-order. Go for a good brand, though even then you can be unlucky. Find out which ones the duplicating companies are currently using - heck you could always buy the blanks from them.
Now the big problem with DIY is making sure that the copies will play on most computers and most stand alone DVD players. You will have to do lots of checks of your copies. From the same tub of 'Verbatim' I have had disks that will play on both, will only play on stand-alones and others that would only play on computers. Weird, but that problem is out there.
If you borrow a mass duplicator tower then don't worry about getting one that has too many drives as the limit on your work flow may well be on how quickly you can print the DVDs. So a 1:4 duplicator may well be fine. As others have said, you will need to buy more than 100 blanks as there will be duff copies, either at the recording or the printing stage. If you duplicate onto DVD-Rs you might need some DVD+Rs in reserve just in case some Parents can't play, or think that they can't play, the DVD-Rs.
Printing: Others have mentioned a number of printers and my experience has been with Epson, (because they were the only 'domestic' machines around when I first needed to print onto DVDs.) I understand that more recent models that others have mentioned may be quicker to print a DVD than the older ones like the R200 and the R300 that I have used. When I first started using these R printers and Verbatim disks I tried to use third party inks and sometimes the Colours worked, but I could never find a Black that stayed Black. (Printing on paper was fine but when printing on the Verbatims the ink faded or didn't 'take'.) From then on I have always used original Epson inks and they are pricey but they have never let me down.
The ink on the disk can be smudged by wet fingers etc. So after printing the disk it is possible to spray a fixer onto the top surface to guard against this. I can't give you any advice on this as I have never done this, but perhaps I should try it.
Cases: Your problem here is the parental expectation on presentation. ie if you provide them with a case with a non-transparent front then you will probably have to go to the extra time & expense of designing and printing an outer wrap for the case - to insert between the clear film outer and the black plastic. It will look good but could you get away with buying clear cases so the label on the DVD can be read through the case?
If you can then there are three choices. The traditional clear plastic DVD case (some are clearer than others though), the CD clear Jewel case or the paper envelope with a see through front. Don't buy the CD Jewel case if the DVDs are to be posted, as they are far more likely to break in the post and thus generate complaints than the traditional DVD cases. Paper envelopes are fine, but parents may feel that they are paying for a proper case.
3. Put the printing of the label and supply of the blanks out to a commercial duplicator. Note I am not including the duplication aspect here, merely the printing and the supply of the blanks.
I was recently quoted £0.89 per DVD (plus VAT and Delivery) for the supply and printing (without duplication) of 400 DVDs (either +R or -R). This will get you quality printing and ink fixing/spraying and you can then do the duplication yourself. However if you divide delivery and VAT costs by 100 then the cost per unit mounts up. You may end up ordering too many, or far worse, fewer than you need.
To sum up your choices are really about minimising the inevitable complaints whilst giving yourself options for the economical supply of the 115th copy should it be needed.
Finally and very very importantly if there is any copyright music etc. included on the DVD, either directly or via the cast singing it, then someone will need to make sure that any necessary permissions and any appropriate fees have been obtained/paid. Permissions obtained for using the music in the live performance may not extend to the selling of copies of the same live performance. I don't know how this works in a school context, so maybe some others can give you some better advice or tell me that I am spouting rubbish!.
Good luck.