Will respite care groups accept my mother's Medicare or Blue Cross/Blue Shield insurance?

Without more information I can't be certain, but what your mom probably has is Medicare with a supplement (BCBS). Unfortunately these do not help with respite care or long term "custodial" care of any kind. Medicare will kick in for a limited number of skilled nursing home days immediately following a hospitalization if the patient meets certain guidelines. One option is to check your local division of aging. Often they have lists of options for respite. Some are unreasonable but sometimes there are volunteers for limited number of hours. If you need a "real vacation" and 24/7 care you can try to hire privately, $150 /day is a good window or check with local nursing homes they often will provide care for 7-14 days.