I don't know about current safe and reputable...I shipped 3 years ago, and companies can go from good to bad in that kind of timeframe.
I can pass on what I learned getting a car shipped from Kansas to Maryland in `09 though...and I would imagine much still holds true when shipping any significant distance (multiple states).
How it works:
Most shipping services you will find online are actually brokers ('middle men') who take the order, then post an online listing to find an independent carrier, or transporter. Transporters 'bid' on the listings they are willing to take.
Pricing:
Pricing depends on the car being operational, a non-op, oversized, if you want an open or closed container shipment, etc. Make sure they know what they are picking up...a driver, a non-runner, a runner with no brakes, runner with e-brake only, etc.
Hint: Tell them JUST what it is! If it's a `69 Roadrunner, it's a `69 Rodarunner. They don't need to know it's a big block, 4spd, A/C'd, bucket seat, fully optioned car. Additional info may have them asking for more money. They have books that give them length/ width/ weight specifications once you specify a make/ model.
Money:
In `09 when I shipped mine, there was no money transfered until a transporter had agreed to pickup the car. At that time, the shipping service charged my credit card half the fee. when the transporter delivered the car I paid the balance. The means of balance payment (credit, cashiers check, cash, etc) vary by transporter because you are paying it directly to them at time of delivery.
Side notes:
- Late spring to summer time is peak shipping season...Costs can be higher.
- Operational cars get shipped more quickly/ easily because they drive on and off.
- Non-op cars can take longer. Since they will need to be winched on and off, they get passed over in favor of the large amount of operating cars available. Depending where they are in the pick-up/ drop-off cycle a transporter might have to winch it more than once or 'restack their load'...not what they want to do, especially in the heat/ rain/ wind.
With my experience I got multiple quotes and did not take the cheapest. I was told it should take 3 weeks, but due to it being a non-op car, it being May(the beginning of the peak shipping season), and my unwillingness to go up in price it took 2 1/2 months. To get the car shipped quicker they WILL call you back and try to get you to: pay more; drop off or pick the car up closer to major interstates in order to make/ save themselves money. I didn't 'need' the car so I held to the original agreement...hence the 2 1/2 month wait. When mine showed up it was three guys who called themselves 'The Big Blue Transport'. the arched trailer had my `66 Belvedere all the way up front (had to be winched on/ off), a (running)50s Dodge pickup with no bed, a Lexus sedan, a Mercedez SUV, and a bike (Victory or Harley)...all on one big open trailer.
I factored in distance (1,500+ miles one-way in my case)/ gas/ food/ loding expenses and determined the transport was cheaper. If it would have been a days drive or less, I probably would have went and picked it up myself. If using some listed online, take the time to read their reviews.
I doubt I answered all questions, but hopefully eliminated some suprises.