A few stabs before owners of good reference books set this straight. 🙂

Firstly, as you likely know, an adverb must come after some verb. Hence, in simple tenses like the present and imparfait, they absolutely have to go after the verb. Thanks to the modal verb, adverbs can often go on either side in this construction, though certain ones may have a bias for one side or the other.

The two possible placements of bientôt are neck and neck in ngrams.

I notice that “déjà” has about the same picture as “bientôt” and so does “désormais”. But “vite” and “prochainement”, while apparently attested (and the examples in book search are indeed comparable uses), statistically seem to prefer the postverbal slot.

On the other hand, the go-to adverb “bien arriver” is far preferable to “arriver bien”, presumably because there’s a difference in meaning: “On va bien arriver” insists on the fact that it’s going to happen, but “On va arriver bien” suggests that it will be done in good health, a rarer utterance… the distribution is similar for “mieux arriver” and “mal arriver”. Meanwhile, “vraiment” appears on either side about equally.

Interestingly, specific times don’t seem to work preverbally, even for a one-word adverb: “On va demain arriver,” “hier arriver”. (Yet “maintenant arriver” is pretty well attested!) A two-word adverbial phrase is utterly out: “On va le matin arriver” marche pas.

Adverbs of place don’t work at all there either: “ici arriver”, “là arriver”…

Incidentally, “poliment dire” is right out:

So is “doucement dire”, so perhaps adverbs of manner are also not flexible…

/terminate guesswork

I think we can say both. In fact, playing and replaying them in my head, I have a hard time deciding what the difference would be. It’s not a difference of language register, and it doesn’t feel like there is any appreciable difference in meaning either.

Curiously, in the very common:

On arrive bientôt

you cannot use any other word order. Ha! It’s possible with the original sentence because the verbal group is long enough! We can also say, on the same pattern:

Nous allons bientôt manger

Nous allons manger bientôt

For other expressions (taken from Luke’s answer):

On va arriver bien

doesn’t sound very natural at all, because the bien is not very clear. I would have said, e.g. On va arriver frais et dispos, which is now clear. This one is a bit tricky, because on va bien arriver can mean in the end, we’ll ge there, the bien being there only for emphasis.

On va demain arriver

is definitely not a possibility. You have to say: On va arriver demain, or on arrive demain, but:

On va doucement lui dire …

On va lui dire doucement …

are both possible. So are:

On va vite arriver

On va arriver vite

with maybe a nuance: the first one meaning that we are going to arrive very soon, whereas in the second, we will arrive quickly (an evanescent or mirage nuance: you hear a nuance at first, but when you really try to definitely put your finger on it, it’s harder than expected…).

A possible rule could be that the verbal group has to be complex enough to make it possible to insert a word before the main part of that verbal group. But even so, there seems to be quite a few exceptions, so it’s not so much a rule as a pre-requisite. If the verb is a simple form such as on arrive, it’s not even possible to contemplate putting a word in between the subject and the verb.

Nous allons bientôt arriver

tends to give equal importance to the action (arriver) and the moment (bientôt) whereas

Nous allons arriver bientôt

is the preferred option if you want to slightly emphasize on the moment (bientôt).

As reflected in the answer to

Quand allez-vous arriver ?

:

=> Nous allons arriver bientôt.

preferred to

=> Nous allons bientôt arriver.

In contrast, Nous allons bientôt arriver is the more idiomatic option when the action is followed by a location adjunct :

Nous allons bientôt arriver au point de rendez-vous. (correct)

Nous allons arriver au point de rendez-vous bientôt. (sounds awkward)

In general, the further away an adverb is from its verb, the less natural the sentence tends to sound.

Il chercha une grande agrafeuse dans le tiroir du bureau de sa
collègue de la comptabilité frénétiquement. (kind of clumsy)

Il chercha frénétiquement une grande agrafeuse dans le tiroir du
bureau de sa collègue de la comptabilité. (better)