You can always apply, but I suspect you won't have much success without either experience or a masters.

If you are not interested/able to get a formal degree then your best bet is to (volunteer for some time, if needed) join a local lab and learn on the job so to speak. Since you seem to have a biology related background it may be easier. I assume you are reasonably comfortable with unix/some scripting since those skills would be needed right away.

PIs usually don't consider hiring someone as a Bioinformatician with no prior experience or relevant education, its a high pressure job around deadlines where the learning curve is pretty steep even for well trained people.

However some larger genome centers have entry level QC positions available which might be a good place to start with next generation sequencing and build your analytical skills from there to look for internal promotion. The pay won't be good but then again you won't be working pro bono either.