FileMaker server should be an option.
But take a look at SeedCode's Gozync tool. I don't think it needs server, though we use it to push/pull data to/from over 300 iPads to server in our organization.
I have worked out a simple method of creating a text file that is composed of the various new records created in the various tables on an iPhone/iPad/laptop/etc. A script places each records data in the text file and then that file is emailed, etc.
The recipient drags the text file into a field and the script then parses out the records among the many tables. This eliminates the many problems caused by connections and other people's buggy software. I'm still debating with myself whether or not this idea is better than the following.
The method is self-controlled and fields can be added to the mother ship without upsetting the children.
One can also hand out the latest version of your user file which is used and then turned back in for the server to import the data. You could email the files to each other. The user can also add new records and then email the file a second time if needed. Importing from a local file is much easier than trying to sync over the internet where dropped connections and phone calls can cause severe problems.
Creating a new work file to email to your users or to have them download shouldn't be too difficult.
Note that you can insert this uptodate file in a container in your server file and the user can then click a button and download that file to their IOS device overwriting the current copy or becoming copy 1, etc. Maybe you could date it such as File 20160122.fmp.
You can also email a copy of the current open file in IOS which I haven't tried a lot but seems to be the easiest and best idea assuming you have someone in house who can reliably handle the import.
The user can also insert their file into a container in the server and the admin can download it for importing, etc. I use this method to bypass iTunes and allow me to transfer a copy of my db from my laptop to my iPhone for field use. It only takes a minute or so to create a copy, insert it in the field and then for my iPhone to click a button and download the file.This works instantly and without flaw when I use my laptop as the host but haven't fully tested it over the internet.
Always make it easy on your users, especially engineers and repairers.