Answer: Yes, that should be allowable, as long as the transportation provided by the shelter is cost-effective. McKinney-Vento allows the LEA to provide or arrange transportation, thereby giving the LEA some discretion to choose the method of transportation, as long as it is appropriate. In this case, if the shelter has the capacity to provide appropriate and cost-effective transportation, the use of Title I funds would be allowable in the same way as it would be to reimburse a parent or provide a school bus. Many areas are struggling with school bus and driver shortages, so this might make excellent sense.
It probably goes without saying, but if the shelter cost is significantly more expensive than other available methods, that likely would be a red flag for an auditor.
A concern to keep in mind would be to ensure the shelter transportation does not stigmatize the children. One of our Board members has horrible stories about being driven to high school in a van that had the name of her shelter in giant letters on the side. Due to that shelter van, everyone knew she lived in the shelter, and she felt a lot of stigma and discrimination as a result. So I would keep that in mind and ensure the transportation avoids stigma.