Answer: Charter schools are subject to the same transportation provisions as any public school, whether they are their own LEA or part of another LEA. They are required to provide school of origin transportation. In addition, for children for whom the charter is not their school of origin, charter schools are required to provide comparable transportation. Those are two separate and different requirements. (The school of origin requirement is in section 722(g)(1)(J)(iii) of the law, and the comparable transportation for non-school of origin transportation is in section 722(g)(4)(A) of the law.)
Whether the charter school generally provides bussing to students does not matter when it comes to school of origin transportation for McKinney-Vento students, because school of origin transportation is a requirement under McKinney-Vento. It is not a comparable service— it is beyond comparable. The specific type of school of origin transportation provided depends on the age of the child and the options available. For older students who can access the city bus safely, bus passes might work perfectly. If parents have vehicles and are able to transport their children, gas assistance is a great option. Taxis, school buses or other arrangements will be necessary in other circumstances.
For charter schools that are part of your LEA, ultimately your LEA is responsible. Contracts and payment arrangements your LEA has with the charter school are outside the purview of McKinney-Vento; under McKinney-Vento, the bottom line is that transportation is provided in compliance with the law, and internal payment arrangements can work however the charter or your district set that up. For those charter schools that are their own LEA, they are responsible. In those cases, your district may split the cost with the charter, but again, those specifics are up to you to work out.