I am wanting to claim two of my kids who live with their mother. With the assistance that she receives(food stamps) would I be able to claim my kids without penalty?

For parents who live apart, the tax benefits for qualifying children can be split. For divorced or separated parents or parents who live apart, the custodial parent, if eligible, or other eligible person who the child lived with for more than half the year, can claim head of household filing status, the credit for child and dependent care expenses, the exclusion for dependent care benefits, and the earned income credit. The non-custodial parent, if allowed by divorce decree or consent of the custodial parent on form 8332 or similar signed statement, can claim the dependency exemption and child tax credit. For post-2008 divorce decrees or agreements, form 8332 or similar signed statement is required. The child tax benefits cannot be split any other way.

If the other parent has agreed to let you claim the dependency exemption, she would need to give you a signed form 8332 to allow you to do that since she is the custodial parent. 

If you are asking whether claiming them would affect the assistance the mother gets such as food stamps, she would have to contact those agencies to determine that.