Depends entirely on the GM. Ask him if he will let your efforts be meaningful or if the AP supports use of a mount.
Definitely do this. It would suck to invest a bunch of feats into mounted combat, then find out you can't actually use your mount.
Assuming a mount's viable, I'd definitely pick one up. Mounted Skirmisher gives you mobile flurrying that's a lot more flexible than what pummeling style offers (no charge limitations, can use weapons other than your fists). I did really enjoy my flurry of nodachi swings sohei when I tried that out.
As for AC, I don't think you'll suffer too much as long as you have a decent dex score. You could always grab the armor expert trait and plan on upgrading to a mithral breastplate once you had the cash for it.
For AC- you can use medium armor, if it is mithral (counts as light for this restriction). And you do not even need to use a feat to get proficiency.
The only penalty for using armor you are not proficient in is to apply the armor check penalty to your attack rolls. If ACP is 0, the penalty is 0. So grabbing mithral breastplate and the armror expert trait works out.
The a mithral breatplate giving you that extra bit, you will have plenty of AC later on. And you will match the turtled up monk early on with just a 'decent' dex (14+4AC armor= 16 dex/16wis). You will generally be on par with rangers and such for AC.
Side note- the eldritch heritage route officially doesn't work due to FAQ. I mean, it was the worse route (since it needed rather large CHA when you already had use for most of the other stats). Still, just thought to mention it. Also, animal ally will give you a mount after 2 feat- it is just you need boon companion to get it up to full druid level. But then again....you do have a bunch of abilities that boost the mount, making it survive easier. You could ignore that third feat.
Anyway, I find that sohei already offers more than enough without the mount (assuming that the feat section is WAY too underdetailed, and as such it is an addition to regular bonus feat options, rather than a replacement- a legitimate argument since the archetype never specifies). With simplified AC, vastly improved attack, and the ability to crit fish with a nodachi or use a reach weapon (great for tactics and improved ability to full attack depending on how you play it)- sohei is fine unmounted.
