Shop supplying equipment to make public healthier (9)
I believe the answer is:
outfitter
'shop supplying equipment' is the definition.
(an outfitter sells specialised equipment)
'make public healthier' is the wordplay.
'make public' becomes 'out' (as in outting someone as gay).
'healthier' becomes 'fitter' ('fit' is associated in meaning with 'healthy').
'out'+'fitter'='OUTFITTER'
'to' acts as a link.
(Other definitions for outfitter that I've seen before include "Supplier" , "Clothier" , "One dealing in clothing" , "Men's clothing shop" , "One supplies gear" .)