To my knowledge, the GDPR does not touch upon this direct issue. It will likely be an issue that must be resolved based on the national laws relating to the right to represent someone else.
Most jurisdictions have rules on how one can be represented by proxy through a power of attorney or similar instrument. One must however separate the issue of designating a proxy from the obligation of a third party to accept a power a attorney. In other words, you might have the legal right to instruct someone else to file a "right to be forgotten" request on your behalf, but the data controller may not be obliged to comply with it unless you file it personally. However, it is possible that the courts will rule that denying a proxy would contradict the intention of the GDPR, and that such a request therefore must be complied with.
But, I guess it would be possible to design a service that assists users with filing multiple requests in the name of the data subject without fronting the intermediary's name. As long as such request is supported by relevant evidence of the identity of the data subject etc. it could be written in such a way that it is not possible for the data controller to tell whether the request originates from the data subject or from someone acting on his or her behalf. The issue would then not be relevant until, e.g., the matter turns into manual correspondence, when the data controller may oppose talking with anyone else but the data subject directly.