There are any number of electronic keyboards, or better yet sample libraries or virtual instruments for PC or Mac (such as the Native Instruments Kontakt environment), which provide the sounds of handbells, tubular bells, carillon bells, or celesta. I'm sure that some digital church organs have these sounds as well.
Here's a computer-based sample library set of handbells with a 5-octave range, the NoiseFirm Handbells. It costs only US$50, but it is designed to work with the full version of Native Instruments Kontakt, whose base price is US$400. (and you need a suitable Mac or PC to run it on, equipped with an audio interface and hooked up to a suitable amplifier and speakers, and of course the musician with the MIDI controller keyboard to play the notes.)
You probably know a church musician who already owns and uses Native Instruments Kontakt and has all the necessary equipment, so it might be cost-effective for your church music department to make a $50 investment in this or another brand of add-on sample library for use by that musician.
I've also found a link to the Bolder Sounds Handbells V2, which appears to offer 6 octaves and far more articulations than the NoiseFirm library, for US $129. It also requires Native Instruments Kontakt.
I have no direct experience with these and could not recommend one over the other.