Do I have to charge HST if I ship goods directly to a US company's Canadian customer on their behalf?

Not necessarily, if the arrangement qualifies under the Excise Tax Act's drop-shipment rules. These rules are built for exactly this kind of situation, a Canadian supplier delivering goods, on behalf of a non-resident business, directly to a third party in Canada, and generally let the Canadian supplier avoid charging GST/HST on that delivery if the non-resident business provides a drop-shipment certificate confirming it isn't registered for GST/HST and that the goods are being held or delivered on its behalf.

The certificate effectively shifts responsibility so that tax isn't charged twice, or missed entirely, as goods move between a non-resident business, a Canadian supplier fulfilling the order, and the ultimate Canadian customer, but it depends on the paperwork actually being obtained and kept, not simply on the general business arrangement looking like drop-shipping. Since the digital economy and platform rules introduced in recent years, whether the non-resident business itself now needs to be registered can also affect how the transaction should be treated, adding another layer to check.

Because getting this wrong can mean HST should have been charged and wasn't, or vice versa, confirm the correct treatment and get the required certificate in place with a tax advisor before relying on drop-shipment treatment for an ongoing fulfillment arrangement.