(See Edit section at the bottom as well.)
For your situation, senior member is correct and is much better than elder member.
If senior is used like an adjective, in situations like this, it usually means that someone has been with the company/site longer, that they have a higher rank, or something else like that. This includes someone who has been using SE longer than you.
However senior can also be a noun. It is a polite way to say old person.
Elder is usually also a noun. It often means one of the oldest people in the group. At other times though, it sometimes means an older member of the group, whose specific job/role is to advise and/or to lead.
You can also say elderly member, which is a polite way to say old member. Old member means the same thing as member who is old. Elder member, however, typically means member who is an elder - which is similar to, but slightly different than member who is old.
- having a higher rank, being with the site/company longer, or something similar
senior member (noun phrase):
member who has a higher rank, has been with the site/company longer, or something similar
- one of the oldest people in the group
- an older member of the group, whose specific job/role is to advise and/or to lead
elder member (noun phrase):
a member who is an elder
elderly member (noun phrase):
a member who is old
As was stated in the comments at one point in time, something closer to chasley - reinstate Monica's answer would be much better for Stack Exchange. Senior member technically works, but Stack Exchange doesn't usually call it that.
Think of a job. Different people have different titles, and the same job might have different titles at different companies. For example, one company might use the word teacher, while another uses the word instructor. Both are technically correct, but each organization has its own preferences.
The same applies for things like senior member. It technically works for Stack Exchange, but it's a little ambiguous; and long-standing member, as @chasly-reinstateMonica suggested, is fairly unambigous and is much more appropriate for this group.