I have an iPhone 4 and I'm borrowing an old iPhone 5s from a family member. I understand that they have two different types/sizes of SIM cards but would I simply ask my carrier (Verizon) to send me a smaller one or would it be more complicated?
The solution to your problem is, either you get it from service provider or cut it by yourself to nano sim size (as stated by other user). You might know that iPhone 4S uses micro sim whereas iPhone 5S uses a nano sim, so to use it with iPhone 4S again after cutting it, all you need is a micro to nano sim adapter which is available online for few bucks. This way you can use it on both devices.
I see two options here – both I have done myself, but maybe things are different here with German operators. So while I don't know if those options would actually work with Verzon, I would still like to recommend them to you:
1) On a similar occasion I asked for a smaller card directly at a service point of my operator. They were able to "transfer" my number from one card to the other basically in front of my eyes. That was very practical, since I didn't have to go through the deactivation/activation process via their service hotline – which can be difficult if the mobile phone is the only phone around.
2) A couple months later I took a special offer: my operator was offering a second card for a one time fee. So I had that second larger/older card sent by post and activated it through their web interface. Now I can basically switch between two devices whenever I want – I just have to login to the web administration interface and point out which card / which device should take incoming calls.