Google Drive is not a server. It is storage in the cloud. You would not be using a single shared copy interactively. I have used Google Drive and DropBox to store my files. I do this as I switch from a Mac computer to a Windows computer. I have switched too quickly, and the copy was not up to date.
You would need to find a service that can host the file(s). Users would access the file from that. You can find reasonably priced options, google "Filemaker Pro Hosting"
1. First of all if you are going to become the "go-to" person for FileMaker for your group I'd recommend getting a good reference book for your version of FileMakers such as FileMaker Pro: The Missing Manaul available through bookstores and Amazon. There are different books for different versions so make sure you match them up.
2. Sharing the "file" (via Google Drive) does not share "access" because the program isn't runing on the Google drive. You can download it from there, but you can't run it from there. Here are some FileMaker help notes on sharing files over a network (http://www.filemaker.com/help/13/fmp/en/html/sharing_data.16.1.html#1029680) from FileMaker.
3. You say your group runs on a " wireless internet connection". Not quite clear what that means. If you mean you high speed internet to a connection point coming into the worksite, that connection connects to a modem and you have a wireless router creating a wireless network - then you are good.
4. It's best if you have a stand alone computer act at the "Host", leave it running in the corner and just let it sit there with FileMaker running. Each of the 5 users then connects (through the wireless network) to the Host and can run the file on their local machine and data updates are saved back to the host. If you don't have a "spare" computer, then one of the 5 people will have to function as the "Host" for the others. This should probably be the one with the most powerful computer as they may see some slowdown if everyone is working in the database at once.
5. To turn on network sharing go File Menu | Sharing | FileMaker Network. That will open the attached dialog box. Set the option to share "On" and you will see an IP Address. That is the address for the Host. The client workstations can then open FileMaker, go File Menu | File Open | Remote and you can add that IP Address as the address for a Host. Once added you will see the available databases that are open on the host listed on the right side. (The application must be open and running on the Host before a client can access it.)
6. IIRC the number of connections from workstations to the Host at any one time is 5 so this should just cover your situation.
Disclaimer: I've never really deployed a working database this way for day-to-day use. However I have used network sharing to allow my co-workers to access a database from my laptop over our work wireless network for testing purposes.
Important small detail: As I understand it, unless one of the 5 users uses the host computer to access the database, you are actually limited to 4 users not 5.