it is known as the drl system (daytime running lights)
daytime lights (headlights) will not come on when vehicle is started. fuse is ok on 1999 GMC Suburban 1500
start with cking pwr at sockets
1 more answer , 2 more comments
Check to see if yours are the type where it has to be in 'drive' for them to come on.I think I've seen some with a set-up like that.
Also with park brake off.
OOPS , I forgot that part yes. thanks to wetry