Samantha, kindle ebooks are mobi format. Your client should be able to send you a mobi file. You should then be able to import it to your kindle device. If you can't figure out how to do that, there are free apps that can import mobi, like Calibre for example.
You can lend books which I am assuming is legal - https://www.amazon.com/gp/help/customer/display.html?tag=skim0x76384-20&nodeId=200549320
My guess is that can only be done with certain books though. This is not my space so I don't know much to comment and not sure if lending is possible, but I assume you aren't supposed to remove the DRMs from the books or anything, so the second half of the 2nd article I sent would probably be illegal, at least in certain countries.
The only other thing I read is to have a shared account -Kindle permits up to 6 Kindle devices to be attached to one account and any number of apps. This would obviously require a lot of trust with personal info though. I don't even know if this works between different countries or if it's just for your family in your house. This one is probably not even an option, at least I wouldn't do it with someone I don't know.
I must be missing something...why wouldn't he just order the books shipped directly to you?
If that won't work, one or both of you is going to have to take a risk or make a compromise.
For example, you could reduce your rate on those jobs to split the cost of shipping.
Or, he could send you a bonus payment that would cover 2-3 books so you could get through the holding period and use that money to purchase these books as necessary.
Also...I'm assuming you've thought of this, but just in case...have you checked your local library?
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