I'm not an aspie, but I am a woman. I have experienced stalking a little bit, one of the things my ex used to sit in his car at the end of the road he knew I was living in and follow me when I left in my car and send me hundreds of pounds worth of unwanted flowers. OP, you're not a stalker, you've bumped into this lady at business events and you are left wondering if there's any mileage left in a relationship. If you'd want to remain friendly in either case you need to tread very carefully. I'd be inclined to buy her something a little special for her birthday, if she's expecting nothing then something very small will be enough, I'd stay away from the romantic types of things, jewellery etc and go for something that you know is personal, a book perhaps? That should be a hint that you still think she's special. You'll also need some words, to make it clear. My advice there, from the aspies that I know, and god knows I'm no expert, is to be very careful with the wording. My partner and his sister are both on the spectrum and I've come to understand there' s a very literal aspect to their understanding. So I could make something clear to them, and unless I expressly tell them that my feelings have changed or modified then the first position I took still stands in their understanding. They don't seem to get it that I, and other people, evolve and mature their opinions and they express surprise when it becomes apparent that mine have. For example I don't recommend that you tell her you'd like a relationship with her again, or apologize for something that's 6 months in the past, I'd probably stick with something like 'I think I forgot what a great person you are' It's non committal enough to work for both scenarios, friendship or dating, but you'd have to take your cue from her replies.
Hope this is of some help