You can see the whistle relay easily with the tender's shell removed. In the link below it is part number WSR-1. The "armature" is the thin piece at the lower left of the picture. There are two electrical contacts facing eacj other (one above the other) one above the other. They stick out beyond the body of the relay. Blow the whistle and watch the contacts move. The lower one will be pulled up until it touches the upper one. When you stop pressing the whistle control button, the armature should drop back. Yours is balky.
Sometimes there is a thin layer of grease or oil that has accumulated on the fixed or moving pieces. Try genly cleaning cleaning them with a piece of thin cardstock soaked with a little mineral spirits. Be sure it evaporates before trying thw whistle again. Sometimes that will fix it.
The main body of the relay is an electromagnet. As lionelsoni says it sometimes retains a little residual magnetism. Slip a small piece ot ordinary tape in so that it keeps the armature separated from the bottom of the magnet by just a hair.
You may need to use a pair of tweezers to get the tape into place. Use a tiny (1/4" square) dot of tape. It should end up positioned with the adhesive side UP adhered to the shiny stationary piece in the center of the relay. Don't let the tape interfere with the movement of the armature. Don't get any adhesive from the tape on the moving part. (You don't want to make things worse by adding some adhesive gunk that would tend to make the armature literally stick to the bottom of the magnet.) Don't bend any parts.Don't put the shell back on until the relay is working smoothly and reliably. Sometimes a second layer of tape is necessary, but not usually.
If all else fails, you may have to consider buying a new relay of the same electricro-mechanical type, or substituting an electronic version.
You may clean the electrical contacts very gently. This is part of regular maintenance and is not related to your specific problem. Do not get any tape or adhesive gunk on the electrical contacts themselves.
Check out this link.
Let us know whether it works. Good luck.