Hi Delores,
You can get away with just having one strap button on the tail block as Dusty says - no need for a strap button on the headstock. You can either tie a bootlace around the headstock and quick release to it - or, even simpler, just throw a bootlace loop tied to the end of your strap around the headstock (works best with scroll headstocks).
It's funny, when I first started playing I used a strap all the time. Then I moved to pads, and now I most often play without anything, particularly when playing the Ed Thomas or Charles Prichard replica dulcimers I have. I think that some of that has to do with the fact that painted poplar wood is just more 'sticky' than a modern super smooth dulcimer but I also think I've just got better at balancing the instrument on my lap as my playing has improved!
A strap is really handy though as it means you can sit on all sorts of objects at various heights and the dulcimer stays put!!!!
Robin