No, it doesn't necessarily have negative connotations. Researching this actually surprised me a bit. None of the dictionaries I've checked mention any negative connotations, it's officially a neutral word.
However, consider the following. (This is so weird, the search results keep changing every time I search). If I search for "sympathiser", I get some interesting results. In (Bing) the sixth result was the Wikipedia article for Nazism. When I entered Wikipedia at the top of the article it said "redirected from Nazi sympathiser". Another time when I tried it was the last result on the second page. Another search I did on Bing showed as second result on the first page an article titled "Terrorism sympathiser released despite tough new laws", and the 8th result was "Broadmeadows murder: Islamic State sympathiser sentenced".
When I switched to Google I saw nothing negative on the first page, but on the second page I got articles about terrorist sympathisers. Just to show you I'm not going mad, here's a screenshot of the third Google result page:
You'll see it's chock-full of articles on terrorist sympathisers, some Islamic extremists, some Irish Republican Army, some Nazi.
In one instance on the first page I got communist sympathisers, but of course it's not there anymore because the internet is trying to make me look crazy.
I mentioned the dictionary definitions were neutral, but it's interesting what Oxford Living Dictionaries actually provides as its main example:
A person who agrees with or supports a
sentiment, opinion, or ideology.
‘a Nazi sympathizer’
Pretty interesting, huh?
It's pretty interesting looking through examples in dictionaries, they are quite often negative, and very often political.
I was quite surprised just how many results I got with negative connotations. The uses of "sympathiser" specifically, by that I mean sympathise inflected with the agentive suffix, seems to very often be politically related.
I'm not sure why this is the case. A guess would be it's because there's often no need to call someone out as a disabled persons sympathiser, or abandoned children sympathiser, or animal cruelty sympathiser, because to sympathise with these things is near universal. My guess is that its use would arise as a rhetorical attack on someone for sharing the values of a group considered disreputable.
However strictly speaking the word is neutral, and its use in your email is not strange in the least, I don't think.