If it is a conventional SID - constructed using ground sited navigation beacons, then you can continue flying with autopilot using your raw data NAV instruments and HDG, (or TRACK in Airbus FBW). We used to fly SIDs out of Heathrow in unsophisticated aircraft using AP and HDG; referring to VOR radials in raw data. As far as I know it is still legal to fly like this - unless certain airports ban it?
One thing that might catch you out could be noise abatement, where less than accurate SID tracking might, in some cases incur a fine to your airline.