Most people can scramble with a good stiff soled pair of walking boots, especially if they have a reasonable edge to them. On lower graded scrambles you won't notice much difference whatever you're wearing, but you could you find yourself soloing on serious ground and realise that it is well worth buying the right kit for the job.
Kit- footwear
There are now a number of companies that supply some top boots for scrambling and scramblers. Lighter than a normal boot they present less for the scrambler to haul up the route, they also offer flexible and well protected ankles for jamming and twisting in cracks.
On the sole is a flattened section stretching from the toe to the inside around the edge that allows the scrambler to stand on small edges with ease and comfort. Some have a slight shank that gives rigidity along the length of the boot and again helps when standing on smaller holds.
The downside with these boots is that they are not particularly water resistant so unless perfect weather is expected, a pair of 'Yeti' style gaiters is recommended for even for summer use - don't forget those deep peaty bogs!
Scarpa have an excellent boot called the El Cap Mescalito and La Sportiva have a couple, the Trango and Trango plus, all highly recommended for scrambling.