Idle-emote all the time, as others have said.Â
Another thing is to avoid placing your character in places that naturally do not inspire walk-ups, especially in bars. It's very easy to fall into this sort of pit when you enter a player made tavern; some are great and has designs with relatively no risks whilst others have some areas that see no natural flow of people beyond smaller groups (which is most likely what they were designed for). Nooks, crannies, and generally the further away from the bar you yet, the more complicated an excuse do people have to conjure up in order to make for an approach. Why would someone go to the farthest, darkest corner when there are seats at the bar still, etc.Â
This is then made even worse by standing about in silence and dropping no hooks. If you're in doubt about how a tavern might have pitfalls, consider observing where people go when they enter the tavern, and how long it takes for someone to fan out towards the edges as the crowd builds up. Â
Also, even though your character may be anti-social and grumpy, they still have to live - certain lifestyles, like that of an adventurer, hints more towards a person being very likely to obtain most of their meals and have their downtime in inns and taverns because if they have a home at all, it might be on the other side of Eorzea. Try to see if you can give them some motivation as to why they would reach out even though they don't like it; maybe wayfinding advice, to check if the food/ale is good before ordering, hearing local rumors to aid them or give them an advantage. Especially the last option is something you can ask literally anyone. Thinking this way has helped me get my own grumpy tall lizardman into more interactions - people are usually fine with him being grumpy, it's not what keeps them away in my experience. If anything, I find that some people may even go "challenge accepted" and figure out why the character is such a grump.
ETA: Another point is also that if you 100% rely on people coming -to- you, then you will always end up waiting a very long time for RP. Walk-ups are intimidating, yes, but what is the worst that can happen? If people flee you probably didn't want to be rping with them anyways.
Another thing is to avoid placing your character in places that naturally do not inspire walk-ups, especially in bars. It's very easy to fall into this sort of pit when you enter a player made tavern; some are great and has designs with relatively no risks whilst others have some areas that see no natural flow of people beyond smaller groups (which is most likely what they were designed for). Nooks, crannies, and generally the further away from the bar you yet, the more complicated an excuse do people have to conjure up in order to make for an approach. Why would someone go to the farthest, darkest corner when there are seats at the bar still, etc.Â
This is then made even worse by standing about in silence and dropping no hooks. If you're in doubt about how a tavern might have pitfalls, consider observing where people go when they enter the tavern, and how long it takes for someone to fan out towards the edges as the crowd builds up. Â
Also, even though your character may be anti-social and grumpy, they still have to live - certain lifestyles, like that of an adventurer, hints more towards a person being very likely to obtain most of their meals and have their downtime in inns and taverns because if they have a home at all, it might be on the other side of Eorzea. Try to see if you can give them some motivation as to why they would reach out even though they don't like it; maybe wayfinding advice, to check if the food/ale is good before ordering, hearing local rumors to aid them or give them an advantage. Especially the last option is something you can ask literally anyone. Thinking this way has helped me get my own grumpy tall lizardman into more interactions - people are usually fine with him being grumpy, it's not what keeps them away in my experience. If anything, I find that some people may even go "challenge accepted" and figure out why the character is such a grump.
ETA: Another point is also that if you 100% rely on people coming -to- you, then you will always end up waiting a very long time for RP. Walk-ups are intimidating, yes, but what is the worst that can happen? If people flee you probably didn't want to be rping with them anyways.