The key word here is indeed beast.
Beast is a defined type of creature within the game, as per page 6 of the Monster Manual;
Beasts are nonhumanoid creatures that are a natural part of the fantasy ecology. Some of them have magical powers, but most are unintelligent and lack any society or language. Beasts include all varieties of ordinary animals, dinosaurs, and giant versions of animals.
Since Wild Shape specifically says
assume the shape of a beast
you are limited to only creatures that have the beast type.
Note that at level 10 a druid of the Circle of the Moon gains the ability to Wild Shape into an air, earth, fire, or water elemental, but it specifically limits it to those 4. This does not allow the druid to Wild Shape into any creature with the Elemental type (such as a Galeb Duhr or Gargoyle).
Wild Shape is pretty important to Moon Druid.
I'm currently playing a Moon Druid, and Wild Shape is key to the effectiveness of a Moon Druid in combat. If your players had wanted a spellslinger Druid, Circle of the Moon was not a great choice. Circle of the Land or another Circle that improves spellcasting would be a much better choice.
TL;DR: Yes, as a Circle of the Moon Druid they should definitely be focusing on their Wild Shapes if they want to be most effective in combat. However, you can play the character and have fun without Wild Shaping in combat by slinging spells (particularly healing and control spells such as Cure Wounds, Healing Word, Entangle, Faerie Fire, etc).
Your best course of action depends on why they aren't using the Wild Shape. Are they making the conscious decision not to, whether for story or personal reasons, or are they just not using it because they're inexperienced and not sure how to use it properly? Or, perhaps they aren't using Wild Shape in combat because they see the party composition and they don't want the entire party to be in the front line swinging at the enemy, so they choose to hang back instead.
Now, if they're complaining about their characters' low HP, then I would hazard a guess that it's because they're inexperienced and don't know any better.
In that case, I'd pull them aside together (with the DM if possible) between sessions and show them some of the cool shapes they have access to (not sure what level you guys are but Giant Hyena, Dire Wolf, Giant Toad, and Giant Spider take the cake at low levels for me).
Yes, you probably have seen a Brown Bear, Lion, Tiger or Dire Wolf at some point.
If they insist that they haven't seen any of those beasts yet, you'll have to talk them through the process of going through their characters' lives BEFORE they became adventurers and consider what they would have seen during that time. Page 24 of Xanathar's Guide to Everything has a really good bunch of tables on it that they should have a look at. The tables are organized by biome (grassland, forests, mountains, etc) and have a fairly long list of beasts up to CR 1 that your characters would have probably seen during their lives pre-adventuring if they were from that type of area.
Discuss with your DM and those two players and see what you can work out. If you have noticed their complaining and lack of Wild Shapes, the DM probably has as well. If you include your DM in the process it'll go much smoother.
Inspiration/Excitement for your players
--Warning: D&D Math Below!!!---
My MVP wild shape pre-level 4 for prolonged combat has got to be Giant Hyena.
I'm going into power-gamer mode for a minute here: doing some basic math, 45HP per wild shape (giant hyena) at 2 transformations per rest is 90HP. If we assume level 3 then you have 4 first level spell slots and 2 second level spell slots, for a total healing amount of 8d8 hit points over the course of 6 turns (one slot per turn), which averages 36 hit points. If you assume that your characters have 12 CON (+1 bonus) and that they take average HP at each level up, they will have 19HP at level 3 on the un-transformed character itself.
Adding this all up, you have 90HP + 36HP + 19HP which is 145HP at level 3, assuming average healing and HP rolls and that you expend all of your resources, and that's without any external assistance such as your cleric or health potions. Show that to your Druid players and hopefully they stop complaining about low HP.
Conclusion
I don't mean to encourage extreme powergaming and it's unrealistic that they would blow all of their spell slots on healing and all their wild shapes in a single combat. I mainly wanted to show how effective the Wild Shapes can make the Moon Druid and hopefully this can inspire and excite your players to get into the next game and try out their Wild Shapes, instead of hanging back and slinging Thornwhips and Ice Knives.
Best Answer
By core rules, likely in none of them (maybe in 1e and 4e)
Below is the perspective on core rules from each edition. It may however be possible that there are further prestige classes and splat books for the various editions that enable it. (Emphasis in quotes mine).
Very likely not in AD&D 1e
Shapechange states (PHB p. 21):
1e had no creature types, but the owlbear (MM, p. 77) description states:
It is not clear if the owlbear counts as a mammal, given there are zootic weirdnesses like the Platypus. 2e, which is very close rules-wise calls it out as a mammal, which would suggest yes.
The other issue is the weight. An adult brown bear, weighs about 500 pounds, but females can be a lithe as 335 pounds, or about 2/3rds of that, so a female owlbear could weigh as little as 850 pounds. By the height and weight table in the DMG (p. 102), the maximum weight for the heaviest type of player character, a male human, would be 175 + 60 pounds, or 235 pounds, and twice that is 470 pounds, far lower. So, unfortuneately there is no way to transform to an adult owlbear. Maybe an owlbear cub.
I strongly suspect that going by Gygaxian sensibilities, it was not intended to be possible.
Not in AD&D 2e
PHB, page 37, Druid class description:
The owlbear in the Monstrous Manual states:
but they certainly are not a real world animal.
Not in 3.0 core rules, but possible with prestige class
The SRD states
The owlbear in 3.0 is a large beast (MM, page 148, at least the printing I have access to). This is differnt a type than animal in 3e, which is its own type (MM, p. 13). As Molot points out, you would be able to do it with a splat book prestige class.
This later upgrades to Large animals and Elementals.
Not in 3.5 (but see options from splat books)
KRyans answer as well as Molot's provide options that allow to do it, for example with the epic feat "Magical Beast Wild Shape" from Complete Divine, p. 90. However, I am focused only on the core rules.
The SRD states:
At higher levels this upgrades to larger and smaller animals, plants, and elementals, no other types.
The owlbear in 3.5 is a magical beast.
Maybe in 4e, for a medium-sized owlbear variant
Page 83 and 84, PHB II:
(Thank you to @KRyan for pointing this section out).
The owlbear in 4e is a large fey beast, so the change would not work for a medium or small sized creature like all PC races in the PHB I and II are. (Thanks go to @Draconis for providing the expertise here -- I've never played 4e myself).
However, owlbear is given as an explicit example for a form to resemble, even though based on size you cannot change into an actual owlbear normally. Maybe if the druid was enlarged magically first? Or maybe you could take the form of an owlbear, but it would be smaller, like an owlbear cub (compare to 1st edition). Medium-sized owlbears are mentioned a few times in 4e material, so it seems quite possible. (Thank you again to @KRyan).
Not in 5e
As you state in the question.