Hi Linda!
Is it possible to just send a letter without a picture and get a response back?
Of course! {thumb2} I
rarely send my own photos (simply because I prefer the photos the celebrties send out) and I've had a perfect success rate, including some rare ones such as Zac Efron who supplied his own photo and signed & personalised it to me (he's a very hard-to-get signer).

So, it is true that sending a photo won't increase or decrease your chances.
There are
some celebrities that don't have their own photos to send to fans, but that is very rare - almost everyone have photos to supply. I know from the top of my head that Christian Bale and Mark Wahlberg only signs photos sent to them - they don't have their own photos to send.
The only time when I would recommend sending your own photos is when you write via venue/movie set. Usually the celebrities don't have anything to sign when they're away so sending a photo would increase your chances on a reply then (although there has been some cases where they
did have photos to sign).
There's only one thing that would truely increase your chances of getting a reply above all the rest, and that is supplying a
Self Addressed Stamped Envelope (SASE).

Obviously you should write a nice letter from your heart too.
Hope this helps!
zanne