There is no such kind of way. Avatars pictures information is not saved in any way, and they must have unified general extension. You can go through the script and change "mbb" to "gif" everywhere you find it, but having .mbb extension is more secure, else if you'd have gif or jpg (there could be some vulnerabilities, having these extensions - I know, for example, that PHP code could be included in JPG file easily).
In general, it is not clear for us, why this extension could be disallowed by your server provider. It is not in the list of known vulnerable extensions, but very custom and related to miniBB only; what about if you upload some specific file to your host for downloading, which will have extension, not in the allowed list? It will be also not available for displaying? If you pay for your hosting, it's up to you to contact your administrator and let him allow this.