I came across a great blog today about an American family that is living and working in Italy with their young daughter.
The blog (La Vita E Belle) covers a wide variety of topics but I found her two posts on Baby Equipment Rentals abroad to be very interesting.
Looking to rent baby gear in Amsterdam? http://lavitaebellablog.com/blog/2013/04/23/how-to-rent-baby-equipment-in-amsterdam/
Stroller rental in Paris? http://lavitaebellablog.com/blog/2012/12/24/renting-a-stroller-in-paris-bambiloo-review/
Not travelling to Amsterdam or Paris and looking for Baby Equipment Rentals in other countries? http://www.babiestravellite.com/baby-equipment-rental-worldwide.html
You can also just Google, “baby equipment rentals ______” fill in the blank with the city/country you are travelling too to find the best match!