If you play Pathfinder or a version of 3.X, the game mastery guide has a number of resources that might be relevant to you. A number of the NPC statblocks represent a number of fun NPC archetypes that could represent employees, customers, or competition. Likely relevant archetypes might include the three city watch npcs (p. 260-261), the pick-pocket and street thug (p. 264-265), the entertainer npcs (p. 272-273), the merchants (p. 284-285), the "road" npcs (. 290-291), the sailor npcs (p. 294-295), the street npcs (p. 300-301), and of course the tavern staff (p. 302-303) and villagers (p. 308-309). You may also find the urban and tavern tool boxes handy too.
IIRC, in the first adventure of Paizo's Second Darkness campaign, they had an adventure where you got involved in running a casino in Riddleport and had a number of... "interesting" games. You might find some inspiration in that if your tavern is a bit on the seedy side. The adventure also included some ideas on Profession checks for running the casino and what the players could do to assist (you might extend these loose guidelines to running an inn?).
Here’s some of my own ideas:
-What does the competition think of the new upstarts? If things get nasty, could sabotage be put on the table?
-This one's a bit more specific and relevant for likely only a single plot hook or two. If the tavern uses rye bread, there's a possibility for an
Ergot infestation. Basically, ergot is a hallucinogenic fungus that is an active ingredient in LSD. Some people think it was what was responsible for the delusions that inspired the Salem witch trials. You could have a similar setup here, with an outbreak of ergot poisonings that are blamed on "supernatural sources". Perhaps people start thinking the inn is cursed, and it's up to the PCs to get to the bottom of this. The infestation may be accidental or a deliberate action as a result of above.