No. The certificate is a statement by an insurance company about a policy it issued, so the insurer transmits it. What you do is put a qualifying policy in force with a carrier that handles filings.
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This trips up drivers who are used to handling their own paperwork with the state. Everything else in a reinstatement, the fee, the forms, the testing, is the driver dealing with the DMV directly. The certificate is the exception, because its entire evidentiary value comes from the fact that an insurer signed it and is on the hook to report a cancellation. A driver-submitted copy proves nothing. Practically, this means the first phone call in a reinstatement is to a licensed producer rather than to the DMV, and the useful question to ask is not "can you file an SR-22" but "which of your carriers will write my record and make the filing".