Verification by check digit, similar to the CPF, is not possible. As already mentioned in the comments, CRM is just a sequential number.
Some time ago, I had made the query through the portal. I would navigate to a URL:
http://portal.cfm.org.br/index.php?medicosNome=mario&medicosUF=ES&medicosCRM=&medicosSituacao=&medicosTipoInscricao=&medicosEspecialidade=&medicosAreaAtuacao=&captcha=wg76pm&buscaEfetuada=true&option=com_medicos#buscaMedicos
And I went through the results table (in html). However, they have now placed a Captcha. It is still possible to do this, but since there was no demand, I did not waste time either.
As you mentioned in the comments too, you tried to use http://www.consultacrm.com.br/index/api
but it is limited to 100 queries.
I would do the following:
Create the interface for the Chm query, and let the user enter the API key, then you have 100 queries per user. In its interface, you can direct the user to the registration and key generation page, making the process easier.