Is it permissible for a sole accountant, whose work has accumulated due to his annual leave, to request additional pay for the hours he works to complete this backlog?

It is impermissible to take anything extra without the permission of the employer. One is not obligated to work overtime unless compensated. If one volunteers to work extra, then it is not permissible to take compensation for it from what has been entrusted to them. One must follow legitimate means and avoid betraying trust, for Allah Almighty says: "O you who have believed, fulfill [all] contracts," and "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]."