If a character is afflicted with Vampirism or Lycanthropy during an adventure, is it legal to continue adventuring with that character while they retain the affliction?
No. Vampires and Lycanthropes are not included in the "Allowed Rules" for character creation or advancement (see page 3 of the Player's Guide). These conditions grant characters powers and abilities that are not suitable for organized play and typically change their alignment to a restricted or prohibited alignment. As such, characters afflicted with vampirism/lycanthropy must have the affliction cured before the start of their next episode or adventure.
Afflicted characters have the following options:
Lycanthropy. These characters can be cured with a remove curse spell. This spell is available as a spellcasting service for 90 gp.
Vampirism. These characters can be cured either by a wish spell cast by a fellow PC (wish is not available as a spellcasting service), or they may be slain and returned to life via raise dead - at the normal cost of 1250 gp.
An afflicted character that exercises neither of those options, or cannot afford them, is retired from play until the character is able or chooses to cure the affliction.