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As an independent contractor, it is legal for the client to require paying for and using their app. If their app is causing you to be late, they cannot punish you for being late though, since it is their own fault and if they seek to punish or terminate anyone for being late, then that person would have a breach of contract claim against the client for their app causing the delays.
However, legally they can make you use the app on their terms as part of the agreement with contractors and the client is liable for the proper operation of the app and any delays caused by the app, not the contractor and they could not legally hold the contractor liable for that.