@Alicia Prokos (and anyone else in the same position) A 401k plan is a type of Qualified Retirement Plan, in which there are - at a minimum - 2 essential roles: Trustee and Administrator.
Plan administrator – The person who is identified in the plan document as having responsibility for running the plan.
Plan trustee – Someone who has the exclusive authority and discretion to manage and control the plan assets.
In the overwhelming majority of plans, it is the business owner and/or the business entity adopting the plan that fill these roles.
Companies that provide 401k administrative services are more aptly called "third-party administrators." This means that the true plan administrator in the eyes of the IRS and DOL remains the business owner/sponsoring business. (These lots more to be said about this, particularly within the ERISA plan context, but this is the gist of it.) The TPA (shorthand for third-party administrator) is simply providing "outsourced" plan services to the plan's true administrator. For ERISA 401k plans (non-Solo plans), the TPA industry is more substantive - many thousands of times over - than for Solo 401k plans
The backbone of a 401k plan is a compliant plan document. To assist businesses in adopting 401k plans, TPAs and other plan service providers "license" use of IRS pre-approved plan documents to their clients.
As mentioned in a prior comment, a 401k plan does not require a financial institution custodian or trustee, unlike an IRA.
The terms "plan administrator" and "plan provider" oftentimes mean different things to different people, particularly in the Solo 401k plan space. Notwithstanding, "plan provider" connotes a service that is primarily focused on 401k document provision and compliance support and "plan administrator" connotes a more involved level of support in management of the plan. However, as these terms are not used "scientifically" in the Solo 401k industry, it's advisable not to focus on those terms; rather, develop a sense of what's required to maintain a Solo 401k plan and identify exactly what is offered by a service provider.
The upshot of all this is that once you've adopted a Solo 401k plan, it doesn't suddenly vanish or become non-compliant because a "plan provider" or "plan administrator" resigns. However, to ensure your plan remains compliant, including plan document maintenance and plan operation, you should engage alternate service providers.