How can I earn enough promotion points to get promoted to E-5 without schools?

Options for schools might be limited but nothing and no regulation says that you cannot flood your command with requests for schools. It never hurts to submit, submit, submit and to write letters to that same effect.

That said, pursue higher education, excel at physical fitness, at weapons qualifications do the best you can as this adds in as well and when given the chance go before soldier boards. Soldier of the month awards you a set number of points as does soldier of the quarter and year etc. Volunteer in your community as this can get you an award worth points. Your OD badge gets you points, strive to be a senior OD.

Get with your command as they can show you a breakdown of a promotion point worksheet and areas that you can work without needing army specific schools that will add up. You are a ways out from even becoming E-4 (P) yet so you have time.

Not too many E-4 have piles of MilEd, but you do have to get SSD and BLC done. Without those, you are dead in the water. I would encourage you to enroll in CivEd and get your military training on your transcript. This step is often overlooked and can get you a leg up as well as several steps closer to earning a degree.
You are in complete control of your PT score. Get to the gym.
Awards are a little more nuanced, but in order to earn one, you have to be in a position to excel. Volunteer for exercises. Seek out additional duties.
You have time, and I like where your head is. Now get after it.

Focus on college education. Although special skills training is important but not a deal breakers when it comes to promotion. However, college gives you more bang for the buck!