Yes, you are allowed to operate a Delaware LLC for any legal business in every state in the USA. Under the US Constitution, your Delaware LLC is viewed as a legal “person” and other states cannot discriminate against this “person” based on where he/she/it was born or incorporated. This is protected under the full faith and credit clause, equal protection clause, and/or substantive due process clause. Additionally, if your company has a physical place of business, employees, assets, or licensing pertaining to another state, then the state where you operate and have significant contacts may require you register there with a Certificate of Authority, letting the headquarter and significant operations states know you are doing business there. This allows you to bring and defend a lawsuit in that state. This registration does not change the incorporation state, it is only a registry of “foreign” or out-of-state companies, in case someone needs to sue your company in its principal place of business.
Agents and Corporations (IncNow) offers this “foreign qualification” service to LLCs in every state and can help you register your Delaware LLC to do business outside Delaware.