Wiktionary lists them as alternate forms of each other.
Merriam Webster's third definition of "whale" as a verb is:
1: lash, thrash
: to strike or hit vigorously
: to defeat soundly
While it offers no comparable definition of "wail".
Etymonline suggests a connection between "whale", meaning
to beat or whip severely
and the noun form of "wale":
a raised line.
In turn, this may be related to the noun weal:
a raised mark on skin.
So, "to whale on", meaning "to lash", definitely has creedence.
Dictionary.com offers only this about "wail on (someone)":
to beat someone. (See also whale the tar out of (so).) : Who are those two guys wailing on Sam?
Again, ambiguity exists as to whale/wail on.
An ngram search shows that "whale on" occurs more frequently than "wail on" and "wale on", respectively.
Then there's this:
One informal meaning of “whale” is “to beat.” Huck Finn says of Pap that “He used to always whale me when he was sober.”
Although the vocalist in a band may wail a song, the drummer whales on the drums; and lead guitarists when they thrash their instruments wildly whale on them.
Although this usage dates back to the 18th century and used to be common in Britain and America, it is now confined mostly to the US, and even there people often mistakenly use “wail” for this meaning.
In answer to your question, both "whale" and the phrasal verb "whale on" have the meaning of "lash viciously". "Wail on", as a phrasal verb exists, but is less frequent than "whale on" and may get you berated.