For each Request Type (Setup > Tickets > Request Type), you can select the 'Use Models' box. This presents the user with the option to choose from either their assigned asset(s) or choose a specific model.
This means the asset must first be assigned to the user, though.
As per Justin's instructions, that is how to enable Asset selection for clients... If you do not have 'Use Models' enabled, you cannot select assets that you have access to whilst submitting a ticket.
You cannot scan bar-codes to assign an asset to a room, there is currently no bar-code scanning functionality in Web Help Desk, however you can implement a way to do this using the RESTfulAPI, although this requires some coding knowledge and time.