Is it possible to make schema changes on a replicated subscriber database? I have a table that is published via transactional replication. A developer has requested modifying the subscribed table to add a column that is to be used for reporting purposes. The column will be updated via an outside process.

It is not prudent to add the column to the published table and insert the data there as the data that is needed to update the column is on the subscriber server.

Is this possible? Is there a better way?
Yes, this can be done. It isn't recommended as if you need to re-initialize the subscription the column will be lost and the values would need to be repopulated.  Can the value be stored as a computed column?  If so, then just add it to the publisher which would then replicate the change to the subscriber.