Well, not the cassette exactly, but I found a video of the group from 1988 at the National Irish Fair in Los Angeles: https://www.youtube.com/watch?v=HduWEJHyUnU. Maybe they played some of the same songs from the cassette? I know it's not the same but it's a start.

Members from the video include:
Richard Gee - Vocals & Guitar
Sue Guiney - Vocals & Guitar
Mark Romano - Mandolin & Vocals
Suzanne Gee - Bodhrán

I also found 3 songs from the album this time:
The Gypsy Rover
Mommy Singing Skibbereen
Barney Rush

The poster, "Aisling Guiney" is probably one of the original members of the group -- you can try posting a comment on the YouTube video and asking her if she could share the rest with you.

Short of cost-based web-stalking services, I couldn't find any *public* contact links for them. BUT, if you search their names on Facebook I was able to find a few of them. Just look for one in California. To be honest I don't feel comfortable giving out their public facebook accounts though, so I'll leave you to find that and message them on your own.