Can I repaint and seal a window sill that has fungus growing out of it?
The wood may give the impression of feeling firm but could very well be beyond recovery. You should try probing the wood with a narrow pointed tool such a flat blade screwdriver. If you can push the tool into the wood any amount then it is a pretty good indication that this needs replacement. 

Replacement of the outside window trim can involve a lot of issues depending upon the type of siding and whether the siding goes under the trim or butts up to the trim. A few more pictures showing the trim installation would help for giving further advice. 

Another thing that will need looking into is to see of the compromised wood extends beyond just the trim. Is the sill and/or frame of the window also starting to go bad? If so this could be an indication for replacement of the whole window.