Is it permissible to request photos of work-related forms from clients due to their unavailability and to avoid writing them out manually?

If filling out the form is an intrinsic part of your job, then it is not permissible to burden the مراجعه (client/applicant) with photocopying it. If you are inconvenienced by this, request an alleviation from your employer. There is no justification for holding the مراجعه (client/applicant) responsible unless they do so willingly and without compulsion from you, giving them the choice between waiting or photocopying. However, if writing is not an intrinsic part of your job, then you have the right to refrain from writing it until it is provided to you in written form.