Can I pay all the taxes associated with being an Independent Companion Sitter?

Actually, as the Employer they should do it. I would talk to a CPA. Ask them how you should handle it.

Yes, you can, but the process might differ by state so be sure to do a browser search for your home state.  Usually it's a simple process where you essentially start operating your business as the sole owner, without needing to file any specific paperwork to establish the business structure. However, you may need to register your business name as an assumed name with the your state's Secretary of State, if your business name is different from your own. You'll also need to obtain the necessary business licenses and permits, and may need to get a Federal Employer Identification Number (EIN) from the IRS.  Please make sure you pay all withholding or employment taxes on time. The fees and interest are hefty if you miss and just creates a headache for you. There are online bookkeeping services that can do this for you, for a fee. I personally think it's worth it so that I never screwed up the amounts owed or missed the deadlines.