In a non-residential cabinet, I have 240v equipment being fed by a single pole breaker and 1 side of a tandem breaker.

Can I tap the hot going to tandem breaker to feed the other side for a 110v outlet.

The main panel breaker would disconnect the cabinet completely do
Isn't this legal.

No. And what you have now is not to code. Install the proper double pole breaker for the 240 line and then you can use the tandem breaker for the 120 line