Jeanne,

It's not uncommon for moving companies to ask for this.

Where you have to be cautious is when you get asked for substantial moving deposits when booking the move. This usually happens when you're dealing with a moving broker, who is someone that connects you to moving companies. You don't want to deal with moving brokers.

These moving brokers often request a substantial amount of money at booking to ensure they get paid by you rather than by the moving company. A reputable moving company won't ask for a big deposit at booking.                                                                                                        

So the $200 deposit they asked of you at booking is reasonable, as is the request to pay up to half the cost of the move at pickup, and half of the cost at delivery. If you're comfortable with your moving company, you should be comfortable with the request.