Which should I write?
He has made several suggestions as to how to improve the situation.
He has made several suggestions how to improve the situation.
I would say several suggestions for improving the situation.
Your form with as to I find grammatical, but a bit stilted; and with just how I find barely grammatical, though I would not be surprised to hear it in informal speech.
I think as to is a form of hedge intended to downplay the "scope" of his proposals. In such contexts, as to (and on) are commonly used to mean concerning, relating to, regarding.
Thus the first version could imply his proposals related to the general subject of improvements. Feasibly some of the proposals concerned, for example, dealing with possible side-effects.
The second implies all his proposals were specifically intended to produce improvements.
I wouldn't go with either. I'd rephrase it, slightly, as one of these, depending upon the verb tense you need:
I also would substitute something more specific for "the situation" unless the context has already made it clear.