Can I keep my domain if I switch web companies?

Every domain has a registrant: the person or company recorded as the holder. That record is what ownership means here. It is not affected by who designed the site, who hosts it, who pays the renewal, or who has the login.

An agency can be listed as the administrative or technical contact and have no claim on the domain at all. It can also be listed as the registrant, which is a different situation entirely, and the two look identical from the outside until somebody checks.

Look the domain up in a public registration record and read the registrant organisation. If it names you or your business, the domain is yours and the rest of this page is procedure. If it names the agency, that is the conversation to have first, before any other work is planned.

Then check you can log in to the registrar directly, not through a dashboard somebody built for you. Access to an account that holds the domain is what turns ownership on paper into ownership you can act on.

While you are there, look at the domain's status codes. A domain showing clientTransferProhibited is locked, which is normal and is set by the registrar, not a sign of anything wrong. It has to be cleared before a transfer will run.

Clear the lock at the current registrar, request the authorisation code, and give that code to the new registrar. The losing registrar then has five calendar days to approve or deny under ICANN's Transfer Policy, and a transfer that is simply left alone completes rather than expires.

Two locks catch people out, and neither is anybody being obstructive. A domain registered or transferred within the last sixty days cannot be transferred again. A change of the registrant's own details can start a further sixty-day clock. Both are policy, both are published, and both are worth checking before a date is promised to anyone.

Nothing about a transfer touches the website. The domain moves; the site stays wherever it is hosted until somebody points the records elsewhere. These are two separate moves that get discussed as one.

Clients register their own domains, in their own name, on their own card, on day one. We take access when we need it and we do not hold the registration.

This is not generosity. A domain held by the agency that built the site is leverage, and leverage is the wrong thing to have in a relationship that is supposed to be renewed by choice. It also makes the answer to this question boring, which is the point.