Hmm, 2.3.6 should be OK to update. Of course they deprecated the Market app so long ago that the Market Updater (I think they called it, before Google Play Services) may no longer even work to update it to Play.
I don't know what else to try. If it was rooted you could see whether there was something you could do (i.e. whether there's a Google apps package that includes Play for that version - take a system backup first!). Without root you may be able to find a version of the Play Store apk at apkmirror.com that's compatible (probably an old one), but I don't know whether you could just install that by itself as a user app and it actually work: I rather suspect it might not (e.g. it may need to be installed as a system app, and at some version it will need Play Services as well to work).