There seem to have only been two major requirements: a unified government and warp-capability. Of course, the citizens of the applicant culture should be able to adapt to the various cultures in the Federation, or they could feel rather left out.
I concur that a unified government probably is a major requirement.
Regarding warp-capability I have strong doubts. It didn't look to me that either Eminiar VII ("A Taste of Armageddon"), Ardana ("The Cloud Minders") or Gideon ("Mark of Gideon") had warp-capability, yet Ardana was a member, Gideon a candidate and Eminiar VII probably, too.
If you can't travel to other places in the Federation, the Federation will provide a transport or come to you. Come to think about it, the Enterprise
picked up the delegates one after one in "Journey to Babel". Theoretically most delegates could have travelled with their own ships to Babel, but maybe that's a procedure in order not to give non-warp members a feeling of inferiority (and they have a chance to get to know each other better, although it looked like some of them already knew each other too "well"