We get receipts from our child care provider. Can we still claim a tax benefit even though they're not licensed?
Licensing wouldn't be related to whether you could use the caregiver's receipts as tax-deductions.
The IRS has a page worth checking out: Top Ten Facts About the Child and Dependent Care Credit, as well as Publication 503: Child and Dependent Care Expenses (PDF). Specifically check out the section "Provider Identification Test" in Publication 503; it's relevant.
The amount you're deducting for child care needs to be for a "qualified person", e.g. a dependent child under age 13, per the IRS, but the caregiver – as far as the IRS is concerned – need not be specially qualified to provide child care. However, the caregiver can't also be a dependent (e.g. your spouse or another child.) See the resources above for details.
Licensing is likely enforced at a municipal or state level, separate from the tax code, and I imagine would be more concerned with the qualifications and conditions of the child care business operator.