Is there any way for an Adventurers League character in the 8th season of AL to gain heavy armor proficiency?

My level 5 wizard has no armor proficiencies, and the only way I found to get it was the following: Multiclass into fighter or barbarian and then gain the Heavily Armored feat.

Are there any other options that do not involve multiclassing?

I still wish to be able to cast spells. My ability scores are around 10 for Cha and Wis, 17 or so for Int, and around 14 for the other scores (Str, Dex, Con).

Multiclass into Cleric

There are certain Cleric Divine Domains that provide Heavy Armor proficiency as a class feature ("Bonus Proficiency"), meaning that you don't need it to be listed on the Multiclass Proficiencies table (PHB, p. 164) - see this question for more information on that.

Bonus Proficiency

When you choose this domain at 1st level, you gain proficiency with heavy armor.

Life Domain (PHB, p. 60) is one such example from the PHB, but there are others (but whether or not you're allowed to pick some of those may depend on what books you're allowed with your "PHB + 1"). Depending on which domain you pick, you might end up with proficiency in martial weapon too.

This also means it would only "cost" you one level multiclassing to gain that armor proficiency, without needing to spend any ASIs on feats like Heavily Armored (unless you need to use the ASI to bump certain ability scores, see below...)

Note that, as per the multiclassing requirement on PHB, p. 163, you will need to have both an Intelligence score of at least 13 (which I assume you will have as a wizard) to multiclass out of wizard, and a Wisdom score of at least 13 to multiclass into cleric...

Also note that your cleric spells would be based on Wisdom, not Intelligence, so unless you happen to have the same Intelligence and Wisdom scores (at time of writing, you have not stated your character's ability scores), your cleric spells would likely be a little less potent than your wizard spells (regarding how high your spell attack bonus and spell save DC are, for example). But a wizard that is capable of healing is always nice to have...

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.

Feat investment

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.

Rebuild your character, if they're below level 5.

There's one very simple method to give your character Heavy Armor Proficiency in a DnD Adventurer's League game, provided that they have not yet played any sessions at level 5 or higher: rebuild them to belong to a race or class that gives access to heavy armor proficiency.

The most recent version of the DnD Adventurer's League Players Guide says the following on the subject:

Character Rebuilding. You may choose to rebuild
your character prior to playing their first adventure as
a 5th-level character (levels 1–4). You can change any
of your character’s statistics but their name. Other,
non-mechanical aspects of your character such as their
alignment, sex, gender, or personality traits can be
changed between sessions regardless of their level.

Your character keeps any rewards and equipment
earned to that point. If you change your characters
class or background, they lose any equipment (along
with the proceeds from selling it, or benefits derived
from it, such as copied spells) associated with the class
or background. Similarly, if you change their faction,
you also lose any renown from their former faction.