Though it will be cool to suggest yourself to be a senior, I would say no. It feels like you are being too greedy in getting a high honor in deviantART; and, if you're new, someone who just draws badly, or doesn't participate in the community at all, then that's worse. Yep, I agree with all the others who said .
A seniority is somewhat more like a honor for being a role model rather than an award for "Most Helpful Deviant" or "Most Talented Artist". Remember, people have different ways of trying to help the community.
But, what if you have helped the community so much that even the community volunteers will feature you in an upcoming journal, and all the other deviants mention you as a helpful person?
I think it's a little absurd to suggest yourself. It's not a position to apply for and it's not like a DD feature. If there was some responsibility or activity that could only be performed by seniors such as with CVs, then it would make more sense to apply. An accolade that represents a form of peer recognition, and which is decided by the community and Fiona is not the sort of thing to request for yourself. What would you say? 'I feel I've done enough to be recognized and as such, I deserve to be suitably honored'? It just sounds weird.
People can suggest themselves if they want, but I think the question would be far more interesting if it was, "what would you think of someone who suggested themselves for Seniority?" considering it's an accolade, not a position...
I agree with most of those who said no. It feels kinda selfish to nominate yourself. Having another person nominate you means that someone believes that you've contributed to the community, which seniority is. And plus, having another person nominate you feels a lot more special since someone else believes that you're a great person!