After going through Reggie's house, documenting all the items that he owned that I also had in my catalog, comparing the prices and doing the math, I've found out that ordering an item through the Happy Home Showcase is 1.2x more expensive.
Here are a few of the items for comparison:
Item Catalog Price HHS Price
Modern Wood Stool 1,480 1,776
Dartboard 1,520 1,824
Exotic Screen 2,250 2,700
In addition, you can only order up to 5 items in a single day. I was able to find additional sources confirming this number here and here, though it's still not stated on Nintendo's official website (which is par for the course for a lot of this game).
So if you want to order something, you'll have to pay 20% more bells, but it may be worth it for those items that you really want and/or haven't been able to find yet. Standard limitations on rare items apply here, so anything you can't order normally through your own Catalog, like the Nintendo set, Gyroids, HHA rewards, etc., you also cannot order through the HHS.