Position of 提供 in 这椅子是专为老年人提供的
It is all because of that 为

Basically the following two sentences have the same meaning:

这椅子是专门提供给老人的

这椅子是专(门)为老人提供的

English translation will be:

We provided this chair specifically for the elders

It is actually quite common to use the 为 to move the verb after the subject to emphasize the "for" 

For example,

My heart only beats for you

我的心只为你跳动