You won’t be able to get behind the wheel for free but there are plenty of other no-cost options when it comes to getting involved.
You could offer to be a Trials driver’s passenger, which means you’ll be in the car during competitive sections but as it’s someone else’s car you won’t necessarily have to contribute any funds.
You could also look at options outside the car such as helping out a team, whether as a mechanic or just a general extra pair of hands.
You could also volunteer as a marshal, helping to make sure events are run safely. With a bit of experience you could move up to officiating or scrutineering, which means checking that competing vehicles comply with regulations.