Am I obligated to provide him with my information?
Yes. But instead of providing your SSN (which I HIGHLY discourage) you should provide them with an EIN (Employer Identification Number) which you can get for absolutely free directly from the IRS in about 10 minutes starting at https://www.irs.gov/businesses/small-businesses-self-employed/apply-for-an-employer-identification-n...
Could I get in trouble (legally) if I ignored his request?
It's perfectly possible. But understand it's not like the IRS will send unmarked black hawk helicopters to hog tie you and haul you back to the IRS dungeons because you didn't follow tax law and report the income. More than likely you would get a bill in the mail for the taxes on that income, plus interest due on those taxes, and possibly an additional fine for having not reported it in the first place. Now if you ignore that bill, I can't offer any guarantees about those black hawks. 🙂
Assuming you don't meet the requirements to be classified as a household employee (I assume you don't since they want to issue a 1099-MISC) you are required to report that income on your own tax return, since it's self-employment income that exceeds $400 for the entire tax year.
If your total income from all sources for the entire tax year is under $12,000 (can be significantly less in some cases) then you don't pay any "regular" taxes. However, for the amount of self-employment income that exceeds $400, you will pay a 15.3% self-employment tax on something close to 97% of that amount that is over $400. So if you've already filed a 2019 tax return or will be filing one, your "late fees" will be minimal. Your self-employment income as a babysitter will be reported on SCH C as a part of your 1040 tax return.
For more details that explains exactly what the SE tax is and more, please see https://www.irs.gov/businesses/small-businesses-self-employed/self-employment-tax-social-security-an...
