Yes, a PC-lycanthrope can multiattack. Under any interpretation some of them do, and I argue that all do. (And you can skip to the last paragraph if you don't need to see the argument.)
Like you said, The PC-lycanthrope gains all of the abilities that don't involve equipment. A very strict reading would say that any action-paragraph that mentions equipment isn't imported: under this reading the wereboar and wererat import Multiattack just fine, but the other lycanthropes all don't import Multiattack.
This reading is too strict to make sense. It has the effect of making some afflicted with lycanthropy faster and able to multiattack (e.g. wereboar), but some not (e.g. werebear). This, even though all monsters with lycanthropy can multiattack.
Contrast this with a reading in which each portion of an action-paragraph is evaluated against "involves equipment."1
The effect of this reading is that the PC-lycanthrope gains natural attacks and the ability to multiattack, but doesn't gain any equipment.
A few examples, all quotes from the 5e MM pp.208-11:
A wereboar gains Tusks, doesn't gain Maul, and Multiattack reads as-is:
Multiattack. The wereboar makes two attacks, only one of which can be with its tusks.
A werebear gains Bite, Claw, doesn't gain Greataxe, and Multiattack can be read like:
Multiattack. In bear form, the werebear makes two claw attacks. In humanoid form, it makes two
greataxe attacks. In hybrid form it can attack like a bear or a humanoid.
A werewolf gains Bite, Claws, doesn't gain Spear, and Multiattack reads like:
Multiattack (Humanoid or Hybrid form only). The werewolf makes two attacks: one with its bite and one with its claws
But even this is a little too much lawyering for me: note that the wereboar or werebear both get to make multiattacks that involve weapons, but the werewolf doesn't. Why? Because we're over-thinking it.
The action is the bold-italic word or few words: the paragraph that follows just explains it. (See also AngryGM's article on abbreviating stat-blocks.2)
Under this philosophy a PC-wereboar gains Tusks and Multiattack. A PC-werebear gains Bite, Claw, and Multiattack. And a PC-werewolf gains Bite, Claws, and Multiattack. I then understand that wereboar can attack with tusks and a weapon, but not twice with tusks. Werewolves can attack twice--one bite and one that somehow uses the hands--but the PC didn't gain a were-spear when afflicted.
1 - Note that for all of the lycanthropes in the MM, the only actions that do mention equipment are those relating to their (artificial) weapon attacks.
2 - Be warned that AngryGM's posts are liberally sprinkled with profanity and/or rude language. Nothing you might not hear on television, but still worth knowing before you innocently head over there.
Player Characters as Lycanthropes
A character who becomes a lycanthrope retains his or her statistics except as specified by lycanthrope type.
Character gains a strength of 15 if his or her score isn't already higher.
There are several references to non-transformed changes in the text around lycanthropy such as:
A werewolf is a savage predator. In its humanoid form a werewolf has heightened senses, a fiery temper, and a tendency to eat rare meat.
As well as in the description of the curse of lycanthropy on page 206
In its natural humanoid form, a creature cursed by lycanthropy appears as its normal self. Over time, however, many lycanthropes acquire features suggestive of their animal form.
The infobox makes it seem that loss of character control only happens if the PC embraces the curse and the DM chooses to take control of the character, and therefore simply not embracing the curse would suffice to keep the character as a lycanthrope complete with powerful boons like damage immunities but without any drawbacks. However, the lore earlier in the lycanthrope chapter suggests that the condition is impossible to resist during full moon:
(from Monster Manual, page 206, under Curse of Lycanthropy)
Even a character that resists the curse is unable to control their bloodlust during full moon, as is traditional to werewolves in fiction. Ask your DM to keep track of the phase of the moon while you're infected so you know how fast you need to get a remedy or other countermeasures.