You can "simulate" renting some warm clothing in Frankfurt.
There is an Oxfam shop in Sachsenhausen, about a 20 - 25 minute walk or about EUR 12 for a taxi from the central train station.
Oxfam is a charity that sells used clothing (among other things) for a very cheap price. Assuming that warmth takes priority over fashion and you are flexible on size, you should be able to pick up a serviceable and warm coat (or jacket or jumper or fleece) for about EUR 8 - 12. With taxi that's a total outlay of EUR 20 - 24.
Because it's a charity shop you have the option of returning your purchase as a charitable contribution before you leave town. This would have the net effect of renting a coat (or jacket or jumper or fleece or etc) for about EUR 20 - 24 (assuming your first leg was by taxi). This is probably better than a real-life rental arrangement.
Frankfurt is very cosmopolitan and I would expect the staff to speak English as do most of the shop staff in Frankfurt.
From the shop to your destination at Roemerplatz, it's another short walk over the "Eiserner Steg" (pedestrian bridge) and you accrue the benefits of experiencing the unique architecture and ambience of Sachsenhausen. Legend has it that the pedestrian bridge is situated near the site where Charlemagne discovered a safe (and eponymous) ford across the Main River and is a popular tourist destination in its own right.
Alternatively, you can take the U-Bahn to these locales and enjoy a commensurate reduction in travel time.
To get a feel for what to expect walking through Sachsenhausen, you can do an image search on the term "mainufer frankfurt winter".
You can get more information about Oxfam from their Wiki Article.