Is one who lies about not having change for customers, under the pretext of needing it for work, committing a sin?

Lying is not permissible. You must be frank with them and tell them that you have change, but you need it for your work. There is no religious impediment in this. You must repent and be careful not to relapse. You can use tawriya (ambiguity) by saying "I don't have [change]" meaning you don't have change in excess of your need, for it is authentically reported from Imran ibn Husayn that he said: "Indeed, in al-ma'arid (euphemisms/ambiguous statements) there is room to avoid lying."