I can't seem to find anything like 3.5e's Ring of Arming (equips/unequips items as a standard action on activation), which is quite useful.
Is there a similar item in 5e or am I doomed to get caught without my heavy armor one of these days?
I think you're missing the spirit of magic items in 5th edition. (or even 3rd, or pathfinder). Just because it's not written down in a book, doesn't mean it doesn't or can't exist. If you are interested in such an item, it's up to you and your GM to devise a way to acquire such an item.
What is different in 5th from previous versions is the idea that you can just amass gold and there will be one of these items magically available for purchase at the local magic shop. 5th edition tries to keep magical items a bit less frequent, and more wondrous and cool when you do find one. The chance that you could just purchase anything outside of spell scrolls or healing potions is reduced significantly.
You don't need it in 5e. You get one "free object interaction" per turn, but you can also spend your action doing a second object interaction. The SRD reads:
If you want to interact with a second object, you need to use your action. Some magic items and other special objects always require an action to use, as stated in their descriptions.
So, you can already spend an action to equip a weapon without need of a magic item.
In 5e, putting it on heavy armor takes 10 minutes. Which is an eternity in most battles. So, yes, you could be caught with your metal pants down.
Sorry, there isn't a replacement for that magical item in the DMG. That doesn't mean you can't ask your DM to homebrew one; but there is no guarantee he'll buy into it.