Where do I get the proper application to get paid to care for my mother?

Elders needs care.
Adults need an income.

Providing your services to your Mother seems like a good solution to both - to her care needs & provide you an income.

Or will it?

Are you wanting to become your Mother's paid support worker? Who will be your employer? Yourself (self-employed?) Mom? An agency?

Although the care provided for one's own elders &/or own children is so valuable, it can be hard to make this financially rewarding.
The reality for many families is the adult woman must leave home for paid work & the elders & children attend daycare for their daytime care needs.

This same question with the same profile was asked about a month ago.
You check with your local social services on aging to see if your state has a program for family members but it may pay very little. Your local agency may have financial support for you to hire caregivers for a limited amount of respit so you can get out of the house.
This is forum of caregivers, there are no forms that you can obtain here

Most children are not paid to take care of their parents. Maybe your mother can afford to pay you from their own funds. If your mother is on Medicaid they may pay for few hours a week at minimum wage.

Here is important info for you to know first:

https://www.agingcare.com/articles/how-to-get-paid-for-being-a-caregiver-135476.htm