Is it possible for someone who is (or really close to) asexual...

To still seek out and participate i'n a romantic relationship?

Yes. Asexual and unromantic aren't the same thing.

As far as I can see, the main problem isn't having the relationship, it's initiating one. Most people seem to ask people out on that first date based on initial physical attraction, and if you never feel that spark it's harder to ask people out.

Most people...almost everyone, actually, make the decision to ask another person out on a date based primarily on that initial physical attraction.

So, yes, if you never feel that 'spark', it's a helluva lot harder to ask people out, because the important first mate selection filter aint working

asexual=/=aromantic, it just means romance is a little more complicated

Yeah. Look up Aromantic on tvt or the other wiki.

Dan Savage actually had some interesting advice on the whole "asexual romance conundrum" deal this week. I'd start by giving the link a click and looking through the first question.

That's about all I can help with at this point.

Does this have to do with a story?

To character creation, most likely.

Yes it does.

I'm going to say that they can.

Based on my own experiences with someone I was once close to, I'm not sure whether he was entirely what you would call "asexual" as he reportedly liked kissing and such. So, there's a difference between being attracted to someone sexually and just finding them physically attractive and pleasant to look at. Both can lead to feelings of affection, and create a desire to pursue someone romantically.