Is Shadow the Hedgehog capable of crying

Yes Shadow is capable of crying, although this has only been seen once within the games, this happened in Sonic Adventure 2 in the Last Story. Amy reminded Shadow about his promise to Maria, which caused a single tear to be shed. The graphics may make it hard to be seen, but it is there.
In Sonic X's adaptation of Sonic Adventure 2, Shadow's tears are easier to be seen. Whilst the series is considered to be non-canon, the Project: Shadow saga do follow the events of Sonic Adventure 2, and as such, Shadow's behaviour overall can be seen as canon.