Certainly it’s possible to start a tiny “venture” firm.
However, as soon as the firm is of any size and has traditional Limited Partners — those LPs will require you to work almost entirely on the fund.
You do see start-up execs raise tiny funds while still working at top tech cos. But these tiny funds are usually their own money plus a tiny bit of outside capital.
If you are going to do it for real … you’re going to have to leave your job.