Unfortunately, game-wise, the character's an adept and so I don't think astral quests are a possibility.
Personally, I wouldn't let this stop you. Game-wise, magic is the ultimate permission to for the GM to wave his or her hand and do what they like.
If you want a canon mechanic for to make them possible, consider the Astral Gateway power (
Street Grimoire, p. 194) or the magical compound Shade (
Chrome Flesh, p. 187), both of which allow anyone to astrally project, even mundanes.
If neither of those are satisfactory, have the astral quest quest be a vision/dream that the character has while sleeping. There's precedence for this in
Harlequin's Back, as (24-year-old spoiler alert!) the entire campaign is an astral quest that the characters have while sleeping.
If astral quests aren't of any interest, then feel free to ignore all this, but if they are intriguing then there are multiple ways to make them a possibility.