I am deaf, a US citizen, and lived in California for several years, and went to Disneyland all the time. So here's some advice for traveling with a deaf family member.
In the US, there is no such thing as an assistance card or special fares for the disabled in the way you are used to. It's very different here and not worth the hassle for a brief visit - particularly since your kids are the ones with the disability, not the adults.
Don't bother with the disabled fares for travel. The majority of bus systems have 2 fares - a full fare, and a discounted fare. Children and disabled pay the same rate. You won't get a "companion fare" for most systems because your child has a disability. Second, every city has a different bus system, so every city has different rules - but in general, the process to get a disabled (adult) pass, is that a doctor licensed in that state has to fill out a multi-page form, sign, and you take it to the bus main office. Then they wait about 2 - 4 weeks, they call the doctor, confirm that she or he actually signed the form, and then mail you a letter (in another 2 - 4 weeks) asking you to come in and get a photo id taken at their main office. So you need a local doctor, and several weeks of waiting time, to get an adult disabled pass. Which, of course, your kids wouldn't qualify for, being children. :)
In the US there is no identification for being disabled, and most places don't have a separate entrance fee for the disabled. Kids generally do have a lower cost entrance fee for museums, movies, etc. It's presumed that an adult will be with the kid, and that adult has to pay full fare. Don't expect to get any special financial considerations because your kids have disabilities.
In the US, most accommodations for the Deaf are for those who use ASL (American Sign Language) and read English. Almost every public museum will have ASL days (where they have ASL interpreters onsite all day long) or you can call ahead (generally 2 - 4 weeks notice) and they will have an ASL interpreter available for your visit. Since you're not from the US, I doubt you all speak ASL! :)
There are a lot of tips for traveling with someone who is Deaf though. Make sure you let the hotels know you have a Deaf family member and that they ensure your room has Closed Captions (CC) on the TV and a light-up smoke detector. Disney has a Closed Caption device you can check out and your kiddo can wear around his/her neck, so when you go into different attractions, the speech is printed (in English) on the handheld device for them to read. Movie theaters all have something similar (but again, in English). There is a Deaf community center in almost every major city, you can email them and ask about particularly Deaf-friendly activities in the city. They aren't all linked, and most are completely non-profit and run by volunteers, but they are great resources. Just Google "Deaf Community center in xx city" and you'll get some results.
I'd suggest you also go onto the TSA's website and download their disability card. It is a blue card you write in the disability of your kid, and the kid carries it with them. When you go through airport screening, the kiddo can hand the card to the screener, and they'll be a bit more careful and patient when they see your child is Deaf.
Finally, the Deaf community is super welcoming and friendly. If you can learn any ASL before your trip, and you see someone signing, feel free to introduce yourself and your kid, and you'll have instant friends almost every time.