No, but even without Legendary Actions the PC will be very seriously overpowered
You already answered yourself in the question - the character gains the vampire's damage resistances, darkvision, traits, and actions". Legendary Actions are not in this list and in Vampire's stat block they are listed under their own separate heading, meaning they are their own separate class of abilities, different from regular Actions.
However, even without that the PC will still get all the following features: Shapechanger, Legendary Resistances, Misty Escape, Regeneration, Spider Climb, a bunch of weaknesses, out of which only Sunlight Hypersensitivity is a mechanically meaningful drawback and the others have impact on role-playing mostly. They also get Vampire's Actions: Multiattack, Unarmed Strike, Bite, Charm and Children of the Night. That is in addition to resistances (necrotic + nonmagical piercing, bludgeoning and slashing), 120 feet of darkvision unless they got that already and 18 in Strength, Dexterity and Constitution. All that is on top of all the abilities they already have since MM states that their game statistics don't change apart from gaining everything mentioned above.
This is going to make the PC seriously overpowered and will unbalance your campaign, hence the common practice of turning the character that becomes a vampire into an NPC. If you want to allow a vampire PC but make it a bit more balanced, the two alternatives I have seen used are either taking Vampire Spawn block instead of a regular Vampire (which will still make the PC pretty strong but not so completely unbalanced), since lore-wise, the character would rise as a vampire spawn rather than an already fully-fledged vampire, or changing the PC's race to Dhampir, which is quite well-balanced and gives the character the same flavour without making them overpowered.
MM in its paragraph about legendary creatures states:
If a creature assumes the form of a legendary creature, such as through a spell, it doesn't gain that form's legendary actions, lair actions, or regional effects.
Given the wording “assumes the form” it is unclear whether this is meant to apply to a character rising as a vampire or only to effects such as Polymorph but I think it doesn’t change anything either way as the rules on PC as vampires list clearly and explicitly all the features that the PC and Legendary Actions are not on this list.