Tyrion definitely earned his in archery — specifically crossbow.
Tyrion — Archery
Requirements: Scouts must identify the parts of the bow and arrow they are using, state range safety rules, and shoot one round with a minimum accuracy requirement.
Tyrion Lannister has earned his badge by hitting a seated target at a distance of several meters. Bonus points for patricide.
HBO / meritbadge.org
Gendry — Rowing
Requirements: Scouts must demonstrate the ability to launch their watercraft, row for 100 yards, backwater for 25 yards, land, and tie a proper mooring knot.
Gendry has earned his badge by rowing since Season 3. Official presentation of badge is dependent upon Gendry actually showing up.
HBO / meritbadge.org
Athletics — The Mountain
Requirements: Scouts must perform a series of athletic tasks within a time frame, including push ups, pull ups, high jumps and running.
The Mountain has earned his badge by... well, just look at him.
HBO / meritbadge.org
Ned Stark — Genealogy
Requirements: Scouts must learn the basics of genealogy and complete a family tree by visiting or using a record-keeping institution or source.
Ned Stark has earned his badge by learning the difference between brown hair and blonde hair.
HBO / meritbadge.org
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If "Game Of Thrones" Characters Earned Boy Scout Merit Badges