You said your mother-in-law lives with you. Does she pay rent, or are you splitting the cost of housing? That would also have to figured into the equation.
If you had a business you would now have to declare the expense on your business taxes. This would also then be income for her, which she would have to account for on her taxes. Remember there are both state and federal taxes involved.
Regarding expenses like diapers. If the MIL had the business she could deduct them as a business expense. If you have the business it would greatly complicate the taxes. Your business would be essentially covering your personal expenses. If your MIL was not a business the cost of diapers would be paid by you regardless of the working situation of you and your spouse.
To claim the tax credit:
You must report the name, address, and taxpayer identification number
(either the social security number, or the employer identification
number) of the care provider on your return. If the care provider is a
tax-exempt organization, you need only report the name and address on
your return. You can use Form W-10 (PDF), Dependent Care Provider's
Identification and Certification, to request this information from the
care provider. If you do not provide information regarding the care
provider, you may still be eligible for the credit if you can show
that you exercised due diligence in attempting to provide the required
The IRS will be looking for an income tax form from your MIL that claims the income.
Getting too cute with the babysitting situation, by starting a business just for the purpose of saving money on taxes could invite an audit. Also it is not as if you just claim 3000 and you are good to go. You can only claim a percentage of the expenses based on the household AGI, the more the make the more you have to have in expenses to get the full 3000 credit, which mil cause more taxes for your MIL. Plus the whole issue with having to pay social security and other taxes on a household employee.
It might be best to skip the risk of the audit. Claiming your MIL as a dependent might just be easier.