Is "The MSO/MSE Split is soon underway" grammatically correct?

The problem is that the word is means "the current state of the thing". It implies temporal information -- now, this instant. It conflicts with the rest of the sentence's meaning.

What is the MSO/MSE split? It's underway.

... But it's not underway now. It will be underway soon. So, it's not underway.

The verb's tense/meaning conflicts with the adjective being used. I think the sentence is still technically grammatical -- it just feels odd because of the construction.

A form that seems to flow better, to me, is "The [split] is happening soon. Please bear..."

This has the added benefit of still making sense when you replace 'soon' with 'right now'.

(as suggested by @badp)

Apart from the fact that the present tense is is perhaps awkward with the future-deictic adverb soon, as commented on in Jeff’s answer, there is a different, more semantic clash going on:

Being underway means that something is not yet here, but it is being worked on, and it is therefore on the way and will be here at some as yet undisclosed time in the (hopefully not too distant) future.

Now, arguably, if you're going to implement a change like this, the work that enables you to say that the change is underway really already starts as soon as it has been definitively decided that the change should and will be implemented (unless you're in the public sector where you'd decide this and then do nothing about it for five years until your department is merged into another department that deals with something completely unrelated to your department and the plans are acrapped altogether, of course).

So it doesn't really make sense to say that the split is (or will be) underway soon—the very fact that this message has been put up is part of the process leading up to the split, which ipso facto means that the split is already underway.

If I were to fix the notice, I would either swap soon underway with coming soon, or I'd just leave out soon altogether and say that the split is underway.

It sounds to me as if the "is" is not the present tense as much as the narrative present. It is less common than (but as valid as) the narrative past tense. It is often used also to describe 'how things are' - general rules or laws of nature or history or society. The sentence

*Ultimatums are exchanged, and declarations of war soon follow*

would be in the narrative present. So your sentence would be OK, I think, if it were part of such a story.

In addition to "will soon be underway", "is soon to be underway" works.

The problem is the disconnect between "is" (present tense) and "soon" (future aspect), which goes away with either of the above constructs ("will" is future tense and "to be" allows for the future aspect reading).

According to “Google NGram Viewer”, soon underway is used almost 4 times more often than underway soon or soon be underway.

The problem is that they are already working on the split now, so it is already underway. For me it should read:

The MSO/MSE Split is underway and will be soon done/here/whatevers.

Not a native speaker though.