My 6-week old child, my husband and I are taking a flight from East-cost to West-cost. Is there a taxi or other transportation company that will provide a car seat on a short notice? And also would you recommend that vs. bringing your own car seat? If the latter, any recommendations for light-weight car-seat that are suitable for such trips?
Many cab companies offer this service, but remember you will need them in both cities, at both ends of your trip.
Some family friendly cab companies are listed at Families Go!. If the cities you are traveling to/from are not listed, you will want to Google what cab companies are available in those cities, and inquire directly with them. Many cities exempt taxis from car seat law, but that may not be an option you are willing to consider (I wouldn't). It means, though, that a lot of cab companies won't bother with the inconvenience.
There are also companies like Traveling Baby Company and Baby's Away that will deliver a carseat to the airport for you. They also supply travel cribs and other baby equipment. You can find alternatives to them by googling "car seat rental."
I have personally lugged a carseat on trips (pre-9/11), and it was a bit exhausting, but doable. I had a canvas duffel bag made to contain the carseat and protect it on the journey. You will obviously be charged for it, and there is a risk that it will be damaged and rendered useless in the luggage hold.
If you purchase a seat on the airplane for your child (usually a fraction of the price for children under 2), the airline may allow you to bring your car seat on board - you would need to consult the airline to confirm this. Some airlines do not allow rear-facing car seats, so be sure to ask that question.
If you decide to get a rental car instead of taxi, many rental car companies also rent car seats.