If a character uses a Ring of the Ram to make an attack, do external modifiers affect the roll? For example, if the character was under the effect of a Bless spell, would they apply that to the roll to hit since the spectral ram's head makes the attack, and not the character?
Bless can't help the Ring of the Ram
The item description says:
The ring produces a spectral ram's head and makes its attack roll with a +7 bonus.
The ring is doing the attack and it has a +7 bonus, that is a flat bonus and what you should roll. Compare this item to the Dancing Sword which says:
The sword uses your attack roll and ability score modifier to damage rolls.
If Ring of the Ram was intended to work this way it would contain this text. It doesn't and can't benefit from bless or your other abilities.
Bless can affect attacks made with Ring of the Ram.
The description for Ring of the Ram includes (emphasis mine):
While wearing the ring, you can use an action to expend 1 to 3 of its charges to attack one creature you can see within 60 feet of you.
YOU are the one making the attack, you are just using the ring to do it. It's the same concept as using a sword, but in this case the ring is the weapon. The bit in the item description about the ram's head is describing how the attack manifests, and doesn't actually mean the ring makes the attack.
And, since Bless says (emphasis mine):
Whenever a target makes an attack roll or a saving throw before the spell ends...
It appears that Bless would indeed affect this particular attack roll.