You have several options you could consider.
First you could talk with your hospital's anesthesiologists and see if they would allow you to come in on cases on your own schedule and see what they do and hopefully participate in some procedures.
Secondly, you can ask a program if they would be willing to allow you to rotate for a very short period of time (one week, etc.).
Once you have that kind of opportunity, you have to be very diligent since you will be under the gun as far as time is concerned. Stay late, do anything extra you have the opportunity to do and hopefully you can parlay that into a letter.
If you cannot do any of the above or any permutation of the above, get good letters from ICU docs, surgery attendings, cardiologists, pulmonologist, or other relevant subspecialties. Those fields have a significant impact on our field in terms of training and understanding so letters from physicians in those fields will wield good influence as well.
Remember that you want great letters regardless of who writes them so even if you can't get an anesthesiologist to write you a letter, get the most bang for your buck with the others. Don't ask someone you are not confident will write you anything less than a stellar letter.