My girlfriend and I are both disabled, and in our mid 30's. We rented a house together with my Mother, and her husband. We're all on the lease. We had a private agreement
To pretty much split the expenses. My girlfriend and I pay the $1000.00 rent, while they pay us a weekly amount of money to cover the utilities, and return a small portion of the rent to us. So far, their share is about $350.00 per month. Far less than their beer budget.
My question is... Is there a legal way for us as tenants to evict them, who are also tenants?

You can try to sue them in court if they are not paying you correctly.

You need to get this agreement in writing so there is no mis understanding

You say you had a "private agreement;" did you get it in writing? Are you saying that they're not fulfilling their end of the agreement they had with you?