Multiclassing and Feats for Heavy Armor
Unfortunately, because you're already at level 5 you can't respec your character. An option at an earlier point may have been to choose a Dwarf for light/medium armor proficiency. But that ship has sailed.
That leaves the options of dipping into another class with Heavy Armor proficiency, but it's unclear what options you have without knowing your stats. You can review this table to determine which options are available.
Based on your stats, you can only multiclass into Fighter to gain the armor proficiency. Your 10 WIS precludes you from multiclassing to Cleric and your 10 CHA means no Paladin either.
However, there are options that don't require multiclassing at all.
First, I want to correct your assumption that you'd need Heavily Armored as a feat if you multiclass. The class itself will provide you with that proficiency, the feat would be redundant.
Again, you've missed an ASI that you can't respec - but you could begin investing in the Armored feats (Lightly, Moderately, Heavily), but your return on that will take awhile.
What's wrong with just Mage Armor?
The primary armor of choice for Wizards is generally mage armor. This gives you 13+Dex AC. That's basically equivalent to a Chain Shirt (medium armor), except you can use your full Dex modifier and not have the limitation of 2. If you end up bumping your Dex via ASIs, you can get up to 18AC at the expense of a level 1 spell slot. Not too shabby.
Why heavy armor?
Heavy Armor itself is tough to get into. You'll need to put points in Strength to even be able to wear it. You'll also be moving around with disadvantage on stealth no matter what.
Ring mail has no strength requirement, and only gives AC14. If you have a +1 Dex, mage armor matches that without even needing to wear anything.
Chain gives you 16AC, but you need 13 Strength for it. And a +3 Dex is equal at that point.
Splint gives you 17AC, but you need 15 STR.
And finally plate will give you 18C, but you also need 15 Strength.
That's a lot of requirements and if you simply pump Dex, mage armor will give you an equivalent without any of the downsides.
A consideration I was making
I'm currently playing a Level 3 wizard with a whopping 9 STR and 11 DEX. It's low enough that my armor choices, including mage armor, don't make a lot of sense to invest in. I'd rather make myself more powerful than harder to hit.
But one thing I've considered, although I don't think I will, is to invest in two feats: Lightly Armored and then Moderately Armored. With Moderately Armored, I can wear half-plate for 15+Dex (getting the two feats would increase my score up to +2). That's 17 AC, but Moderately Armored also gives shield proficiency, so I could also wear a shield boosting me to 19AC.
Alternatively, simply spending two ASIs for DEX would give me AC 15 mage armor. So this seems like a potential positive and a better option than opting for Heavy Armor - especially since I didn't have the strength requirement anyway.