Do Orcs make good Clerics?

I recently remembered the Orc Eye of Gruumsh from the 2014 Monster Manual. I noticed that all of its spells are Cleric spells, and that gave me an idea for an Orc War Domain Cleric character. The problem is that since Orcs are basically designed to be martial classes, I'm not sure how suitable the Cleric class is for an Orc. I know picking Protector for the Divine Order feature is an option for a martial-inclined character, but I've heard people say how bad Protector is compared to Thaumaturge. Should I go through with the idea?

The impact of species choice on class choice in 5.5e (and even 5e, with TCoE rules for ability scores) is pretty minimal, unless you're really into hyper-optimization of game mechanics. Any species can be a good Cleric.

Protector is not spectacular, but it's fine. The issue with it is that Clerics don't get access to Weapon Mastery features, and (aside from War Clerics, a limited number of times) can't generally make more than one weapon attack in a turn. This means that you will still generally be dealing more damage with cantrips than with weapon attacks, especially if you take Potent Spellcasting at level 7. But damage output isn't everything, especially not for a Cleric, and the difference isn't all that huge; the heavy armor training alone might be worth taking Protector.

If your goal is to be a front-line smashy smashy Cleric, another approach (especially if you're starting at a higher level than 1) is to take a level of Fighter first, then multiclass into Cleric. One level of Fighter at the bottom gets you everything that Protector does — so you can take Thaumaturge and thus effectively have both options — as well as a Fighting Style feat and Weapon Mastery.

I think Orc is a great species for Cleric, perhaps one of the best (if we're talking 2024 species).

Many Clerics don't have much to do with their Bonus Action (now that Spiritual Weapon requires concentration), so Orc's Adrenaline Rush allows them to turn it into 2X mobility and some extra HP at the same time. This is of course very helpful for reaching your melee targets, but it also pairs really well with Spirit Guardians, allowing you to hit most of the battlefield.

Of course, if you're going War Domain, there would be some competition for your Bonus Action, but it could be good to have both options, since they are both limited use. And Orc's second survivability feature, Relentless Endurance, is also of course great for frontline combat. 

Not sure I'd say Protector is bad, especially if you're planning on a martial build. Quite the opposite: Heavy Armor & Martial weapon helps you get the most out of melee combat. I think many often go with Thaumaturge with non-martial Clerics for the extra cantrip and a little more skill, since you can still get pretty decent AC with Cleric's Medium Armor & a decent DEX score of 14. You should definitely feel good about Protector if you are going for a melee Cleric, I'd say.

What level is your campaign reaching?

In the unlikely event you reach level 20, you may wish to go Protectorate and not take the Fighter dip at the beginning. Getting access to wish, even if it is 2d4 long rests of a cooldown can be very powerful. …at least in my opinion. Can’t say I’ve ever been in a position to use the wish spell.

Then again, you get stuff sooner if you start Fighter, but then you put of your other Cleric abilities by a level. Is it worth it? I’m very curious, especially since I’m considering purchasing Arcane Unleashed and would want to make an Arcana Domain Cleric build to have just in case.

In 2024 rules, orc is an excellent cleric. Adrenaline surge isn't a big deal, but one of the most important jobs of a cleric is to stay up (as long as the cleric is up, the rest of the party can go down and you can still recover), and Relentless Endurance is made for staying up.