So I decided that it would be fun to start an artifact deck, and was wondering if I could use Coat of Arms to give +1/+1 to an Artifact Vehicle and an Artifact Creature Construct if they were the only artifacts/creatures on the battlefield?
Artifact isn't a creature type, and Vehicle is an artifact type not a creature type, so Coat of Arms has no effect on a Vehicle and Construct. You'd need multiple Constructs.
You can find the full list of creature types in the rules (205.3m, reproduced here), but a creature's creature types are always printed after "Creature" on the type line, never before, and will never include things like Artifact, Enchantment, or Land - those are types, not creature types. (And outside of some corner cases, everything after "Creature" on the type line is a creature type, assuming you're looking at the Oracle text in Gatherer or a sufficiently recent printing.)
(Technically creature types are subtypes, but for convenience they're referred to as creature types.)
Coat of Arms works based on creature type, the subtypes of a creature card. A cards type is broken into three parts:
- Supertype (like Legendary or Basic)
- Type (like Creature or Artifact)
- Subtype (like Construct or Vehicle)
These are laid out like this on the card:
Supertype(s) Type(s) - Subtype(s)
Artifact and creature are both types, construct and vehicle are both subtypes. Not all subtypes are treated the same however, some are creature types, some are land types and others are artifact types. Construct is a creature type while vehicle is an artifact type. This means that multiple constructs will power each other up, though multiple vehicles will not, as they have no creature types to trigger Coat of Arms.
The rules on card types are covered by the 200 section in the comprehensive rules, subtypes are 205.3 and creatures are covered by 205.3m. Anything not listed in 205.3m will not be counted for Coat of Arms.