Probably ShipStation is your best option. As it integrates directly with ClickFunnels, and from Shipstation you can integrate it again with any 3rd party fulfillment and tracking solution, In fact, it can be quite handy even if you are handling shipments yourself.
ShipStation is really my #1 recommendation to handling orders from Clickfunnels, though the only limitation (or issue) is It doesn't support fulfilling from outside the US. (unfortunately) thus it's only for domestic shipments handling within the US.
Hope ShipStation team, change their mind anytime soon and realize the huge amount of potential (and revenue) they are leaving on the table.
So as a sum up, if you are shipping from within the US, no matter what kind of warehousing or fulfillment solution are you using, Then ShipStation is your tool.
If you are Shipping from abroad, or Dropshipping, then I would go for Zapier.
Though I've seen some complaints about missed orders when using Zapier, but that's really your only choice. Unless you are planning to build your own backend software or do the orders handling manually.