It’s the hour at which they usually go out for their daily walk
It’s the hour they usually go out for their daily walk
Depends on what you mean by "need". Your second sentence is perfectly understandable, but it is not complete.
Normally preposition + which can be replaced by a preposition at the end of a sentence:
It's the thing for which I'm looking
is equivalent to
It's the thing I'm looking for
In your case, this "transformation" would turn your first sentence into
It’s the hour they usually go out for their daily walk at
but that becomes so long it's quite unwieldy.
A shorter sentence such as
It's the hour at which they eat
could be turned into
It's the hour they eat at
but again, that's a peculiar sentence.
So why not replace "at which" with "when":
It's they hour when they eat
Or, in your seentence:
It’s the hour when they usually go out for their daily walk
I don’t find your second sentence to be perfectly understandable. It can be read in both of these ways:
It’s the hour (at which) they usually go out for their daily walk.
It’s the hour(,) they usually go out for their daily walk.
One thing with English is it has no rules (only conventions) and it is spoken worldwide in many dialects and with many differing conventions. So you can’t read the mind of your reader. Even a writer in Boston and a reader in San Francisco may make dramatically different assumptions. But your reader may be much further away than that. Therefore I recommend you write what you mean to say. Err on the side of being explicit. The exception to this would be poetry. But if you are writing in order to be understood, then I recommend you say “the hour at which.”