No, it's probably not the motor, it's most likely leaves in the fan cage.
I pulled the fan out of my Mums '99 Civic at the weekend and cleared out a
ton of leaves all the way from the air inlet above the fan to the pollen
filter. You can access some of Honda's workshop manuals from here and see
how it all goes together, choose the Civic 1995-1997 it's got the same
internals as your 1999 model:

http://www.honda.co.uk/car/owner/workshop.html

In a nutshell: you remove two screws at the hinge or the glovebox and put
that to one side and then unclip the power connector to the fan (underneath
the passengers right foot), unscrew 3 screws (8mm socket required DON'T use
power tools or you'll strip the plastic!!) the fan can then be dropped out
with a bit of wriggling to clear the trim panel in the footwell and cleaned
out. Then remove 4 screws (again 3 require an 8mm socket 1 needs a 10mm
socket) holding the metal bar at the base of the glovebox, drop one end to
the floor and you can then remove the access panel to the pollen filter and
clean out the duct from the fan to that point. Put it all back together and
check you have no spare parts left over and it'll be quiet again. You may
need to replace both the pollen filter and/or the resistor pack (it controls
the fan speed) the resistor pack sits immediately above the fan and is held
in place with two screws just pull it and visually check the 3 coils to see
if they are burnt or brittle; change it if it looks bad.

Chris