In Critical Role campaign 1, my friend is convinced she remembers Keyleth (moon circle druid) using Revivify at some point. Revivify is not a Druid spell, but she swears up an down she remembers Keyleth casting it.

According to the character sheets, Keyleth ended as a level 20 Druid (no multiclass) and did not have the Magic Initiate feat or other feats that might grant her spells from other classes. Critical Role is a d&d 5e game that was converted to 5e from Pathfinder, so perhaps this was a hold over from Pathfinder?

Can a Circle of the Moon Druid cast Revivify without using 5e feats that grant additional spell list options? Any known Critical Role house rules that would allow this would also be helpful as that is the origin of this question, but also any 5e or Pathfinder RAW would be good

No, they cannot

Revivify is a 3rd level spell. The only way to gain access to a 3rd level spell that is not part of your class spell list is to multiclass. (The Magic Initiate Feat can get you access to a 1st level spell and cantrips, but no higher).

Keyleth...

The scene your friend is most likely thinking of is from Chapter 1, Episode 3 where...

Grog was healed from having his Int reduced to 0 by an Intellect Devourer.

After figuring out what happened...

Keyleth casts Greater Restoration, using the effect of ending "Any reduction to one of the target's ability scores."

As part of the houserule set that is maintained for Critical Role, this spell is a Challenge

Here is how Matt runs the world. He feels that some of the problems in some RPGs is that death becomes flippant when magic allows you to completely undo it. In his campaigns certain powerful spells like greater restoration, resurrection, and things like that are a challenge.

Source

I would note that it is not possible that Revivify was cast here. Between the party member going down and the spell being cast, a Long Rest was taken. This is far too long of a period, as Revivify can only be cast within 1 minute of a character's death.

No, a Circle of the Moon Druid cannot access Revivify from their spell list

The Circle of the Moon Druid does not gain any additional spells from their circle, and Revivify is not on the Druid spell list.

Some Magic Items do give access to spells from other spell lists. Notably a ring of Spell Storing would enable a spellcaster to cast any spell from any spell list that was cast into it.

Keyleth is accessing a number of non-druid spells via magic items, but Revivify is not one of them

According to CritRoleStats, who track all things Critical Role, Keyleth has never cast Revivify.

The Third Level spells Keyleth did cast during the campaign, in order of # of casts are:

• Fireball (via Spire of Conflux) 10


• Dispel Magic 9


• Call Lightning 8


• Tidal Wave 7


• Wind Wall 7


• Daylight 6


• Protection from Energy 5


• Speak with Plants 5


• Water Breathing 4


• Plant Growth 3


• Fireball 2


• Lightning Bolt 2


• Sleet Storm 2

The real outlier in that list is Lightning Bolt, however it's possible that she got that spell from one of her magic items (in a similar way to Fireball being accessed using the Spire of the Conflux).

Looking through the various sheets of the Google Sheet source it appears that Keyleth cast Lighting Bold in Episodes 7 & 24. Episode 24 is listed as being from Diplomacy (a hombrewed magic item ). The Episode 7 listing doesn't have a source, but I would expect it to also be from a magic item of some sort.

Source: CritRoleStats Campaign 1 General Stats Page, specifically the All Spells Cast google sheet linked on that page.