My heart goes out to you and your family in this situation, dealing with someone with dementia is always tough, but you all now have an added layer of problems. Does your MIL have a Durable Power of Attorney naming someone as her agent? The agent can work on obtaining these documents on her behalf if she is not cooperative. If there is not a POA, you may consider filing for Guardianship/Conservatorship, which might be necessary to complete some of the paperwork ahead. You should seek a local attorney to help you determine if this is the necessary course of action. (Elder Law attorneys in your area can be found at https://www.naela.org
-- then click on "About" then "Consumers" and "Find an Attorney".
The agent or Guardian can call her insurance companies, utility companies, doctor's office, etc. to request these documents on MIL's behalf. You can likely order a birth certificate through the state or county where she was born. This may not require any forms of ID just to order, and many states have order forms that you can download online. Once you have a birth certificate and a utility bill, for example, you may be able to get a state issued ID card.
Good luck to you and your family!