Is it common to say There is some sort of reason to support my stance on this? How do we say some sort of reason in such sentences?
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Some sort of is generally used to when the speaker doesn't know the exact nature of whatever is being referred to. For example:
implies that we don't know exactly what sort of trouble he was in.
Therefore your sentence:
implies that you are not entirely sure of the reason(s).
If this is not the case then write simply:
Both of these remove the implication that you are uncertain what these reasons may be.
As Shoe notes, "some sort of" indicates uncertainty. It seems unlikely in this context, like you're saying, "I have reasons, but I don't know what they are." (Well, I guess a lot of people engage in muddled thinking like that.)
What exactly are you trying to say? If you simply want to say that you have reasons, then leave out the "some sort of", as in, "I have reasons for my stand on this." You might want to give some indication of what those reasons are, like "The scientific evidence backs up my stand on this" or "The accounting department agrees that my stand on this will make them more efficient" or whatever the context is.