I'm in a party of 5 others + me, with a rogue, ranger, paladin, warlock, and fighter.
I love being a Circle of the Moon druid, but also I feel the obligation to fill the healer role (a bit of pressure from the party, but I don't mind too much because druid is my favorite class).
Can I participate in combat and still be a good healer?
First, healers are not mandatory. There are other ways of handling the situation that don't require a dedicated healer spot.
Second, a circle of the moon druid is a fine healer, as long as everyone can wait until after the fight for healing (which is a good thing to shoot for anyway). You still have all the same casting slots that any other druid does, but you also have shapeshifting. The more time you spend in combat shapeshifted, the less reason you have to burn spell slots on attack spells (and thus the more you can spare for after-battle healing).
Third, the Moon Druid excels at being a HP sponge that doesn't himself require healing. Your damage output in beast form is going to be on the weak side, compared to the full-melee classes, but every attack that it soaks up is an attack that you won't have to spend resources healing later - all that damage just falls off when you shift back.
You'll be fine.
You should be fine because healing HP during combat is rarely optimal except to revive a fallen ally. Sure, in those cases the Druid's Healing Word is nice because it's ranged and doesn't take an action, but providing Goodberries beforehand (allowing party members to revive an ally if needed) is almost as good.
Also note: you are actually not the only healer in your party (since both your Ranger and Paladin have healing spells). Plus (since Goodberries last 24 hours) both you and your Ranger can create Goodberries the night before an adventure utilizing your unused spells, allowing for plenty of healing between combats the next day.
Ben Barden's answer has already covered the healing from Circle of the Moon druids specifically, so I'll expand on healing for that party composition in general.
I'm currently playing a campaign with an almost-identical party (Sorcerer instead of Warlock); Our Druid (Circle of Twilight) usually avoids healing spells except in the most dire of situations, due to the character's personality. However, as long as they position intelligently, this isn't an issue for the rest of the party - Both of the frontline classes, Paladin and Fighter, have healing abilities, with the Paladin's Lay on Hands being particularly important for burst healing. Meanwhile, the Ranger's Healing Spirit is crucial for drawn-out fights, though even more location-dependant - While many DMs will limit its out-of-combat usage (10d6/character is, frankly, broken), in-combat it can be used both to counteract damage and spot-heal downed players. Add Cure Wounds, potions of healing, and short rests, and you should have no trouble with sustainability.
The most important thing for this party is going to be positioning - The primary source of burst healing, the Paladin, can only heal at Touch range, which means if your Rogue gets caught while scouting ahead or your spellcasters flanked, you are going to have serious trouble, especially as Healing Spirit (which can only be moved 30 feet/round) should usually be adjacent to the most dangerous enemy (to allow your melee fighters to rotate through it without provoking attacks of opportunity).
