In theory, yes, you could drop "of", but you would also have to drop "the". In practice, it is unlikely to be used that way.
The reason is that "beginning" can be used in two ways. In "the beginning of", "beginning" is a gerund: a verb form being used as a noun. Nouns can have articles ("the"), and can have phrases like "what they will do" attached with "of".
The other way for "beginning" to be used is as a verb, in the present continuous form "is beginning". The object of a verb is not marked with "of", so that's why you can leave "of" out. However, you must also leave out "the", because verbs don't use articles.
So, "this is beginning what they will do" is grammatically correct, but it is awkward phrasing. A more natural, idiomatic way to use "is beginning" is with an infinitive ("to"-verb) as its object, as in the following examples… or, of course, to turn it into a gerund followed by "of".
This is beginning to do what they will do.
This is beginning to demonstrate what they will do.
Also, the meaning of "this is beginning what they will do" is strange. It means that "this" (some object?) starts to do something that "they" (some group of people?) will do later on. I would not expect "this" to be the right pronoun here. I would want "they" to be the subject of both verbs ("begin" and "do"), or else for there to be different verbs:
They are beginning what they will do. [Still a bit awkward.]
I am beginning what they will finish. [Good grammar. A bit overdramatic though!]