I know the rules for Texas Hold ‘Em very well, but I recently played a tournament that has raised a question. Here it is:

I was playing a tourney at a local bowling alley and got down to 3 players. I was on the small blind. I pushed
all in and the big blind called. I won the hand and the
person on the big blind was eliminated. From here, it
seems simple, I have the button and there is a dead small blind, but the other player that was left in claimed
that I had to post a small blind for a second time in a
row. Is this right and if so, is there some website or rule
book where I can verify this?

No. The SB is dead. You took the player out who would have been that. There is only a BB. I’m sure you have seen that many times playing online. Unless of course you are heads up.