Can I run civilization 5 with a Integrated ATI Radeon™ HD 3100

Yes, though it'll probably work with the lowest settings.

256 MB ATI HD2600 XT or better,
256 MB nVidia 7900 GS or better,
or Core i3 or better integrated graphics.

I think it would struggle, if possible a cheap upgrade would help.

it will but it wont be fun to play, i recently installed it on a 3200HD and at the lowest settings it is very laggy, id think it would be even worse after exploring a lot of the map, also the visuals is so bad after turning them all low that id rather play civ 3

edit: im also playing civ 5 on a desktop with 9600gt 512, it still lags when i had explored more than half of a large map
which is why im no considering to get something at the range of gts450 to gtx460/6850