How can I prove elder abuse?

I agree with funkygrandma. Ask the police to do a wellness check.

Make an appointment with an elder attorney. They will be able to instruct you on the next step.

Best wishes to you and your mom.

Find an elder law attorney who practices in her state and invest in a consult. That's the only way to know for certain what evidence you will need and if your case has a chance. I would for certain ask an attorney if you should call APS right now to report her as a vulnerable adult, or maybe apply for emergency guardianship... maybe better to not give your sister any hint of what you're doing so that she can't showboat for APS, the police, etc. I wish you the best outcome for your mom.

Does your mother have dementia so severe that she cannot say who she wishes to live with? Does your Mom own the property on which she pays mortgage and lot rend? And does the sister live with her and care for her? How have you learned about the abuse (sister falling asleep, who pays for what and etc.) How do you know your Sister is preventing your Mother from calling you.
If your Sister is as dangerous as you claim then you will have to have witnesses and proof because any fight in court over guardianship of a NON-COMPETENT (proven and written documentation) will be very costly (10s of thousands). And if you lose you will be responsible for court costs.
As your sister is the CHOSEN POA at this time she has a big step up in this siblings war. And to tell you the truth, her job is a difficult one; I am not certain why you are so wanting to take it on. I can't imagine your Mom chosing the "evil sibling" as her POA, so that's where I am starting from. I CAN imagine a Mom on the phone complaining about the family member she lives with, because that sort of thing is often what we do, sadly.
The fact that your Mother's complaints are currently so miniscule as "I am lonely; she's always busy" doesn't bode well for any court fight. It doesn't constitute abuse. If you suspect abuse you are best to open a case with APS in your area and turning over you evidence of abuse, then asking if you should apply for temporary guardianship over your mother.

I guess you can always call the local police dept. and ask them to do a wellness check, saying you haven't heard from your mom in a while, and that you're worried. You can also call Adult Protective Services and share with them your concern. They will then have to investigate the situation. Good luck.