If they are of different degrees, then they are automatically linearly independent over the field, so you may assume that your elements are all of the same degree.
>> restart
>> K = ZZ/32003
>>
>> R = K[x,y,z,w]
>> I = ideal(x-y,w-z)
>> B = R/I
>>
>> someEltsInB = {x,y,z,w}
>> degOfEltsInB = 1
>>
>> J = ideal(someEltsInB)
>>
>> M = module(J)
>>
>> leadCoefficient hilbertSeries(M, Order=>degOfEltsInB + 1)
I'm sure there are better ways to do this but this works.