I know of no reason why you would need the restaurant's permission. However, if you advertise using their name, you may need their permission.
If you operate under an assumed name, you need to register with the county (if a partnership or sole propreitorship. If you enploy someone there are a number of legal requirements, so I will assume that you will have no employee. I am not aware of any other legal requirements other than a driver's license and proper insurance if you use an automobile.
You may want to consider forming an LLC or corporation, and you should consider whether more than the minimm of insurance is adequate.
This answer is for discussion purposes only and will not be considered legal advice. Further, a court could potentially decide the question contrary to my answer.
Set yourself up as a delivery service hired by the end consumer, see an attorney, set up a limited liability entity, determine your share of the profits and losses, if you are going to have employees. And what coverages you need.
To the PROSPECTIVE client, please call myself or another attorney for your choice with more detaiils and an appointment. My PRELIMINARY answer to your question(s) is for general purposes and based upon what little information you have conveyed. It is based on such limited information that the general answer should never be relied as a reason for your action or inaction. My response does NOT establish an attorney-client relationship and such may only be established by mutual agreement, and the signing of a written retainer agreement, which will generally require payment for our services, as this is what we do for a living and, just like you, we must get paid for our work.. .
I agree with Thomas and John. You need to start an entity and file for an FEIN. I can assist for a fee. Sincerely, Reese Serra
This advise is only being offered as a courtesy. We do not have an attorney-client relationship and I do not represent you. Additional, the advice I have provided was based on the limited fact pattern provided by you. If the facts differ, my advice may not be correct. If you live in Michigan and would like legal assistance with your issue, feel free to contact me: 586-615-0756 / Reese8484@aol.com (ADVERTISING MATERIAL)