Does a lawyer that is retained by another party have the right to withdraw and seek monies from client? If possible, why?

A client can terminate his relationship with a lawyer at any time, and for any reason. The lawyer has the same right, so long as the termination does not prejudice his client's legal position. That generally means the client has adequate time to find other counsel to represent him or her. A lawyer's decision to terminate the relationship, however, does not mean that the lawyer waives his right to a fee. In general, the lawyer would be entitled to the reasonable value of the services provided to the time the relationship ended. That value is not necessarily equal to the hourly fees that the attorney has billed, but the amount of those fees might be relevant to a court's determination of what is reasonable.

Unless your agreement with them does not allow for it, the answer is yes. Just like anyone else, lawyers should be paid for services rendered. If a contractor builds 50% of a house you would not argue that they are not entitled to any money, would you?