As a 91B how do I go on patrols and get extra combat training?

You can forget that as a 91B. In country you might be assigned convoy escort or maybe become part of a security team. Your primary mission is to keep equipment rolling so others can do their mission.

Better question to ask:
How do you, a 91B, get more 91B training to be a better 91B, which is the purpose to your existence in the military. No one needs you to know how to kick in doors. They need you to be a good 91B.
When vehicles start getting deadlined for electrical system issues, "but I know how to do a patrol" is going to be the most meaningless set of words to ever leave your mouth.

I was a Sailor, and an officer at that... pretty much the LAST guy one would expect to be convoying though the KG Pass behind a 240G... yet there I was, for nearly a year. Some things come down to fate... a lot comes down to choices. In the Navy, we say, "choose your rate, choose your fate". I'm sure if they let everyone have exactly the MOS they wanted, most people would want to be some kind of Infantry... the Army does a lot more than Infantry. If that's your passion, I'd suggest that you seek every opportunity to get into a combat arms MOS and excel there-it's not easy, but certainly "easier" for you than many. I knew a guy who gave up his commission in the Guard to get the MOS he wanted on AD as an enlisted man. I've heard of people going back through Basic after having earned significant rank during prior enlistments. Where there's a will... there's often a way.

The best way to get out and do combat oriented missions while also being a mechanic is the be on a recovery team with a H8 Wheeled Vehicle Recovery ASI or to be on the contact team in your unit.

1975, 2nd 505th Inf 82nd ABN out of Fort Bragg

Don't know how you'll be able to do it now a days, but, here is a picture of two MP's attached with us in the jungles of Panama.
Their mission was to support us 11B, with the securing of hostels that were captured. We did take care of them very good, and had them carry some of our ammo, for M-60 and M-79, and they were 82nd MP's at Bragg. They also, jumped with us into Panama.

In picture, are the two MP's standing at left. By the way, there were no captured hostels while in Panama for 3 mouths, 2nd 505th Inf didn't take captives.

As a 91B the only way that is happening is if you pass RASP and join the 75th. But if you’re high speed the Ranger Regiment is always looking for volunteers

Be in a unit that would do those tasks.
I used to take a mechanic on every mission if they were available.

If you, as a mechanic get pulled to go on a raid and kick in doors.....man, some serious issues with manning took place. You want to kick in doors and go on combat patrols, then you need to switch to a combat MOS that does those things. Or go SF. The odds of you going and kicking in a door are probably about the same as winning the Powerball lottery. Well, maybe a tad better than the Powerball odds.

I would say ask your COC first to see if there would be an opportunity for you to cross train.
Also, as others have stated look for a change in MOS or ASI, I had the opportunity to pick up 38B Civil Affairs. It was certainly would fit your wants. However, I must tell you, Be Careful What You Ask For!!! It is not fun and games outside the wire.

Join the Infantry