Keep in mind that while you don’t know them, they also don’t know you. If you aren’t reaching out, that won’t change.
Their challenge is 1 new person. Your challenge is much larger, of course, a lot of new people. This imbalance is the very thing that can make it harder to for the 1 newbie to start integrating with the group. There are so many people for the newbie to recognize and know, whereas they are only learning 1 new person.
I wouldn’t decide right away that the barn culture is ‘unfriendly’, just busy. Understandable, most people there likely have busy lives and barn time is on a schedule. They are there to focus on a their tasks and finish by a certain time.
Lessons might be a short-cut to meeting people. But depending on what you want out of your barn time, I don’t think it is the only way. I would also keep low expectations because sometimes a lesson group doesn’t actually know each other. They lesson, then go back to their time scheduled tasks.
As you start to recognize a few people, you’ll get a feel for who is more talkative and who isn’t. Also, others who are also just boarding there and not doing lessons. You will have a bit more in common with the other non-lesson boarders, to get started on self-introductions.
You can start finding reasons to introduce yourself briefly “Hi I’m Jane, I just moved in a few weeks ago with my gelding, and I just thought I’d meet a few people. Your chestnut mare is so pretty!” With a smile – smiles break the ice and signal goodwill. Very useful with new people. If you aren’t a habitual smiler, practice!
Only until you are settled in with some barn people you are enjoying and are accepting of your usual demeanor, whatever that is.
Given the busy nature of the barn, I’d keep first introductory remarks to a brief, general observation. They can give you their name and a brief remark of their own and carry on what they are doing. If they smile when they give their name, it’s a sign they are friendly enough.
That is, you can introduce yourself in passing and not appear to be sidetracking someone by trying to initiate a full conversation. Especially while they are task-oriented. It’s also good if you smile, break off the contact and keep walking on your tasks, to let them know that you don’t intend to be distracting, and you also have your things to do.
I have to note that this kind of self-introduction would go well in the culture in my part of the country. There are other areas/parts of the world that are culturally less friendly and less inclusive. Where many people really don’t want to expand the group of people they know. They may be more abrupt and less welcoming. Just keep working your way through the group, and you’ll find people who are more interested in other people.
Especially with a larger group that has less frequent contact, be ready to later remind people you meet of your name (and which is your horse). You may also want to keep some secret notes on your cell phone to help you remember names and which horse goes with who.