Conversion to fixed will be a bit of a challenge; conversion to single speed will be much easier.
For SS, if you have an MTB rear wheel in reasonable nick just get a suitable rear splined cog and a spacer kit (On-One, DMR, Charliethebikemonger etc) and a tensioner (or look up the bodges to use your old derailleur as a tensioner).
For fixed with vertical dropouts, the issue is tensioning the chain. You *cannot* do this with a chain tensioner (because bad things will happen) so you're into a new rear wheel (or serious bodging of the old one) plus all sort of other arcania involved half-links, magic gears, strong QR's, filed dropouts and frequent fettling.
If you're determined to go fixed, much better to look around for an older frame with horizontal dropouts instead.