I would like to write my son's name, James, phonetically in Klingon script. From my reading there isn't an "ay" sound in the Klingon language as the a makes an "ah" sound. Should I drop the "a" altogether and just write it as "jms"?
If you want it to match the sounds you're looking for, go with jeymS. It doesn't fit standard Klingon phonology, but it's the closest you'll get to "Jay-mes".
If you want to go with canon, then Emeads suggestion of jemS is what you have. This is more the way it's said in TOS.
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I believe the "ay" sounds you are looking for is actually "ey" in Klingon transliterated using (can't remember the word for it right now) the non-pIqaD script. It looks like this: 
in pIqaD.
Note… I'm not a fluent Klingon speaker or any kind of expert, but I think "jemS" is the way James T. Kirk is spoken and written according to the canon (and knowing James T. Kirk is important to the language — look it up if you don't recognize it ;). So, if you want to go with that as a precedent, I would use that.. even though it is pronounced something like: "jehmsh" The whole name (jemS) looks like this: 
(sorry about the spacing/formatting… I had trouble getting it uploaded. The last letter should be more in line with the first three)