Check your particular bike. All of my horizontal drop old road bikes have a little steel screw built into the dropouts. Super easy to replace if they're missing, check your frame. Not a tensioner but it will allow you adjust how deep or shallow you place the wheel in the frame. Probably enough to make up for the slack in running a single speed without a tensioner.

If you want your chain tighter, grab something soft like a tennis ball and cram it between the frame and seat-tube, then tighten your axle nuts/quick release. It's imprecise but mostly works ok.