If I purchase a phone in the US is a European charging cord compatible?

I think the cord is the least of your worries rand (and the cord probably won't be a problem). Buying an iPhone in one country to use in another country can have many disadvantages:

Make sure it will work in your country! --> LTE Find the right iPhone for the right country or region. - http://www.apple.com/iphone/LTE/

and https://en.wikipedia.org/wiki/List_of_LTE_networks

Also check if your intended carrier requires a CDMA or GSM phone.

Some carriers will not accept unlocked phones - https://discussions.apple.com/message/20359060#20359060 - E.g., "Virgin Mobile, like all CDMA carriers, will only activate devices on their network that were initially sold for use on their network." and https://discussions.apple.com/message/33206605#message33206605 "The iPhone 7 and 7 Plus sold unlocked and for use on AT&T and T-Mobile is not physically compatible with the Verizon and Sprint networks."

Discounted phones sold in some US locations may be locked to a carrier. You have to complete the contract with the carrier before they will unlock the phone. In the end you are not saving money and may have to wait a long time. https://discussions.apple.com/thread/252444302?answerId=254609062022#254609062022

Other issues - https://discussions.apple.com/message/33176952#message33176952 - "The Japanese carriers are notoriously difficult to work with when it comes to unlocking. If you are not a customer of the carrier it's locked to, they will not unlock it."

Warranty restriction! - http://www.apple.com/legal/warranty/products/ios-warranty-document-us.html - "IMPORTANT RESTRICTION FOR iPHONE, iPAD AND APPLE TV SERVICE. Apple may restrict warranty service for iPhone, iPad and Apple TV to the country where Apple or its Authorized Distributors originally sold the device." <-- The warranty does not outright deny warranty coverage in other countries but there are many limitations to the effect Apple almost certainly will refuse to do repairs under warranty or even repairs at all except in the country of purchase. The device would have to be sent to a friend in the country of purchase to handle any repair claims because Apple will not handle international shipping.

Devices may also have tax added to the cost (e.g., in the USA prices do not include taxes), and you may have to pay import duty when bringing an expensive device into your country.

For vacation purposes there shouldn't be any issue but if you are truly moving there to live I would buy in Europe.