You can install a lot of LDS Apps on your Kindle Fire (which is an Android device) by selecting APPS; STORE; search for "LDS." KINDLE Touch is probably Android also.
On a regular (non-android) Kindle it is not easy. You can go to lds.org on your home computer and print a membership list (print-to-file) as DOC or TXT or PDF type file then send it as an attachment in an email (with no subject or message data) to
yourkindlename@kindle.com.
Yourkindlename will be different. To find out what it is: select HOME; MENU; SETTINGS; Page 2 has instructions for Send-to-Kindle E-mail and displays yourkindlename.