A number of solutions.
There are a number of software solutions that can help you do this. However, given the mobile nature of the people involved, I suspect that the ideal solution for you would be both cloud and mobile.
Mobile solutions will allow the captains or the crew to take a photo of the receipt / expense and submit it to you. You can break such solutions into two types: (1) services that require the end users to take the photo and do the data entry on their phone (such as iXpenseIt) and (2) services that automatically extract the data for you from the photo (such as Expensify or Receipt Bank - the company that I work for).
Consider storing the records online.
In theory destroying older records should be fine. However, to be safe, you could suggest to your client that he scans the records and stores them online so that there is no cost to storage.
There are a number of services that enable you to do this including: Evernote, Dropbox or Receipt Bank (my company) where we have a free DIY plan if you just want to store the items.
What accounting system do you currently use?
There are a number of solutions available that specialise in this, including Shoeboxed and Receipt Bank (my company) or, for very large numbers of employees, Webexpenses.
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