While it's impossible to say for sure if the next Avatar will be able to master all four elements for some reason or another, it seems unlikely that what Amon did will be cause of any difficulties.
Firstly, Amon's ability comes from
bloodbending, and was a physical severing of an individual's connection to their chi energy. The next Avatar would have a new body and their chi-connection would be intact.
Korra regains her ability to bend the other three elements at the end of the first season, when her previous incarnation Aang appears to her in a vision and restores them, and in the process teaching her how to restore others. If even Korra herself can continue to use her mastery of all four elements after what Amon did to her, there's no reason to assume the next Avatar would have trouble from that.