Can Bitcoin blocks be organized into shards?
If you mean whether block verification can be distributed so that one entity can participate without storing the entire blockchain, this isn't compatible with the current design, but maybe a new design can do that. Also, suggestions such as this aim to improve the ability of lightweight (SPV) clients, who do not store the entire blockchain, to verify transactions concerning them specifically, without changing the core protocol.

If you mean whether a single entity can distribute the blockchain among several machines on the same LAN, then this is possible. The basic operation needed for block verification is to look up the details of a previous output, so the natural sharding key would be output hashes. One could also cache the newest outputs which are the most likely to be used in the following transactions.