Military Hunger Prevention Act This bill requires the Department of Defense (DOD) to pay a basic needs allowance to certain low-income members of the Armed Forces. Specifically, those eligible for the basic needs allowance are members who (1) have completed initial entry training, (2) have a gross household income that did not exceed an amount equal to 130% of the federal poverty guidelines of the Department of Health and Human Services (HHS), and (3) do not elect to decline the allowance. In situations where a household contains two or more eligible members, the bill specifies that only one allowance may be paid per year to one member they jointly elect. The amount of the allowance must be the aggregate amount equal to 130% of the HHS federal poverty guidelines minus the gross household income of the covered member during the preceding year divided by 12. The Defense Finance and Accounting Service (DFAS) of DOD must notify, in writing, individuals determined to be eligible for the allowance. The notice must include information regarding financial management and assistance programs administered by DOD for which the individuals are also eligible. Individuals who seek to receive the allowance must submit specified information to DFAS.