Unfortunately, yes - you must re-register with every texting vendor, because your registration is specific to sending texts via that vendor's specific texting lane. Thankfully, the Impactive registration process is quite straightforward - read more about it here.
Note that if you are an electoral entity re-registering with Impactive after already having obtained a Campaign Verify token to register with a prior vendor, you can use the same prior Campaign Verify token for your Impactive registration. Put another way, that Campaign Verify step does not have to be re-done - only the Impactive registration does.