I play a mobile game that has a number of Russians in my guild. I can type out what they say in Google Translate, but on my phone I only have the Latin A-Z available. I know I could go get a Russian keyboard app, but is there a way to communicate in an elementary fashion with Russians using just the Latin alphabet? Is there a mapping of Russian letters to Latin letters that would allow me to type to them?
What you are searching for is translit:
Translit is a method of encoding Cyrillic letters with Latin ones. [...]
The translit system arose when Russian speakers first needed to write their language on computers that did not support the Cyrillic script.
Basically what it means is that you will write russian words using the latin keyboard. There are certain rules however (refer to Common Transliterations in translit table), which can be a bit tedious to write and to read / understand too (although they are common and well-known):
odin || один chetyre || четыре shest' || шесть sem' || семь
It's possible to convert russian words to translit and back with some services, like translit.net. That can be done only when words are already written in Russian though; you will need to know respective rules of transliteration to write word only with Latin keyboard.
You can add a secondary keyboard layout in Windows and do the same on your phone.
You can buy Cyrillic stickers for the keys from eBay (though once you start you get used to the letter positions and won't need them after a while). It takes practice and is difficult at first, но это стоит того.
Google has "Input tools" which allow you to type in multiple languages using latin letters (including Russian):
Please clarify your needs.
Do you want to be able to respond to them using a Latin keyboard? Just type Russian words in transliteration, like "Spasibo". I do it in all the games that have problems with Russian layout.
Do you want to type what they say in Google Translate so it translates things for you? Unfortunately, it can't accept transliteration as far as I know, but instead of typing you could use some application that can detect text and translate it for you. I am not a smartphone user, but I know that such applications exist.