Has it worked OK before?
Try inserting a 15A fuse in fuse position E (marked as 15A Quad lamp) and/or a 20A fuse in fuse position F (marked as Spare).
I think that with some models there's a Front Control Module used to control the high beam and also the DRL and perhaps the relay is there, but that doesn't explain the fuse box cover marking in your vehicle or the missing relay connector
Here's the wiring diagram for a 1998 Dodge Ram there is no headlamp relay used.
The low beam is fed via a 15A fuse #1 in the passenger compartment fuse block (marked as headlamp switch) and the high beam is fed via a 40A fuse #5 (marked as headlamp switch) in the engine bay fuse block. Their operation is controlled by the headlamp switch and the dimmer switch only
(click on image and when it opens click on it again)
Here's a link showing the fuse block layout for a 1998-2001 Dodge Ram.
The fuse location description matches the wiring diagram above and the layout matches the one shown in your image BUT the designations don't match.
It shows 5 (40A) the Headlamp switch (the wiring diagram shows Fuse 5 40A) but the designation shows FA5 brake lamp (this is assuming FA5 =fuse #5) and your image shows 20 Amp Stop Lamp.
This link for a 1998-2001 fuse box layout shows the headlamp switch as Fuse F and the high beam headlamps as Fuse E and the high beam relay is R4. Your cover shows Fuse E as 15A quad lamp and Fuse F as Spare and R4 as Quad Hi Beam relay.
The two layout links above are for the same year models and the layout matches your cover but the functions for the fuses etc regarding the Hi Beam headlights aren't the same. Very confusing