The sentence is actually not entirely syntactically sound, and it's also stylistically poor, because its length makes it difficult to parse.
Consider a far more simple sentence:
✔ It is as necessary for me to sleep as it is for me to eat.
In short, for me, sleeping is just as necessary as eating.
Now add two more instances of as:
✔ It is as necessary for me to sleep as much as possible as it is for me to eat as much as possible..
In short, for me, sleeping as much as possible is just as necessary as eating as much as possible.
But it's really only the as at the start of the sentence and the as at the end of the sentence that are required for this type of construction.
We can easily remove the second or fourth. (Or both, which would result in the original example sentence):
✔ It is as necessary for me to sleep as it is for me to eat as much as possible..
The sentence in the question can be analyzed in the same way.
First simplify it:
✔ It is as necessary for me to be vigorous in condemning [some] conditions as it is for me to condemn riots.
Hopefully that makes sense.
Now add in the rest:
? It is as necessary for me to be as vigorous in condemning the conditions which cause persons to feel that they must engage in riotous activities as it is for me to condemn riots.
The construction is essentially valid, except that the second as doesn't work. It seems to be an addition that's unnecessarily attempting to serve the same function.
Look at the phrase on its own, with the as removed:
✔ [It is necessary] for me to be as vigorous in condemning the conditions which cause persons to feel that they must engage in riotous activities
The phrase only makes sense without that as. It doesn't make any sense when it's there.
As such, it's a mistake.
The sentence should really be this:
✔ It is as necessary for me to be [] vigorous in condemning the conditions which cause persons to feel that they must engage in riotous activities as it is for me to condemn riots.
That's only two instances of as.
The sentence is still unnecessarily long-winded and difficult to parse (it should be rephrased), but it's now syntactically valid.
Note that the second as, which I analyze as a mistake, might make sense if it follows from a previous sentence. But without that context, there's no way of knowing.