Cairhien was in the midst of a brutal civil war, with the Damodred House one of the main players, so they didn't have troops to spare even before the Shaido came (not sure whether Morgase knew that before she decided to go to Amadicia). More importantly, Rahvin sent Andor troops in Morgase's name to press her claim for the Sun Throne and thus put them against the Damodred forces, who were one of the main claimants, so Morgase couldn't expect any support from them.

Anyway, if Morgase wanted to overthrow Gaebril, her only realistic option was contacting the Houses which later on were the first to support Elayne - Dyelin's Taravin and the 4 with teenage High Seats (Mantear, etc) and get the men from the Trakand estates. and also try to get some of older Queen's Guards on her side. A foreign army of any kind supporting her was a bad choice given the extreme patriotism and pride of the Andorans - it would've guaranteed the whole realm uniting against her, and none of the neighbours is in the same league as Andor military.