Two cleric domains get proficiency with martial weapons in DnD 5e, and dwarves have a racial proficiency

Cleric Domain

PHB p. 62 and 63 identify Tempest and War Domains as having martial weapon proficiency(for those two domains, using identical language). Other domains do not have proficiency with martial weapons.

Bonus Proficiencies
At 1st level, you gain proficiency with martial weapons and heavy armor.

Race

Likewise, a dwarf cleric of any domain will have racial proficiency with a warhammer.

Dwarven Combat Training
You have proficiency with the battleaxe, handaxe, light hammer*, and warhammer (PHB. p. 20)

Feats

IF the DM allows feats/variant human at first level (optional rule) ... (credit to @nitsua60 for reminding)

A human (variant) cleric could also grab the Weapon Master feat at creation and elect warhammer as one of the four proficiencies.

*errata change from throwing hammer to light hammer (credit to @MichałPolitowski for reminding of errata change)

Different AC calculations don't stack; you pick which one to use

As Slagmoth pointed out, this general topic has been asked and answered many times by the designers, and an official answer appears in the Sage Advice Compendium:

These [AC calculation] methods—along with any others that give you a formula for calculating your AC—are mutually exclusive; you can benefit from only one at a time. If you have access to more than one, you pick which one to use. For example, if you’re a sorcerer/monk, you can use either Unarmored Defense or Draconic Resilience, not both. Similarly, a druid/barbarian who transforms into a beast form that has natural armor can use either the beast’s natural armor or Unarmored Defense (you aren’t considered to be wearing armor with natural armor).

(The entry goes on to clarify that AC bonuses, like those provided by shields, do stack with AC calculations; it's simply that if your features give you different ways to calculate your AC, you pick one.)

In addition, Keith Baker - original Eberron designer, and primary author of the Wayfinder's Guide to Eberron - has posted an FAQ on his website about the Wayfinder's Guide (which he has continued to update to answer questions). It contains a section about the Integrated Protection feature, which specifically answers this question and references the Sage Advice ruling:

If I’m playing a warforged barbarian, do I get to add my Constitution to my Integrated Protection? What about a warforged monk?

No and no. Integrated Protection and Unarmored Defense are two separate features that set your AC, and the Sage Advice ruling on this is that you pick one—you don’t combine them. If you have your Integrated Protection mode set to darkwood core, you are considered to be unarmored and can thus use Unarmored Defense—but you’re always either using one or the other to determine your AC.

The section also clarifies whether the Integrated Protection feature counts as "armor"/"being armored" for other features. It's a worthwhile read.

In short: you have the choice between two different AC calculations:

• The warforged's Integrated Protection: 11 + your Dexterity modifier (+ your proficiency bonus, if proficient with light armor)

or

• The barbarian's Unarmored Defense: 10 + your Dexterity modifier + your Constitution modifier

They don't stack. You simply have the choice between one and the other (presumably you'd choose the one that gives you a higher overall AC).