Austin Healy Bugeye Sprite
MG Midget
MGB
Triumph TR 4, TR5, or TR6
Rule #1 - buy a car that has already been converted for racing. Building one yourself will cost twice as much as buying one.
Cars above are: Cheap to buy. Affordable to run. Lots of parts available and parts are cheap (unlike early 911 or 912 parts which have skyrocketed). Lots of expert advice available on the internet. Fun to drive. Low power so you probably won't kill yourself.
Good reading:
How to choose a car for vintage racing | Hemmings Daily
11 Best Buys for Vintage Racing | Hagerty