pandas divide row value by aggregated sum with a condition set by other cell

try this:

In [79]: df.assign(ID_new=df.ID/df.groupby('Source').ID.transform('sum'))
Out[79]:
   Source  ID    ID_new
0       1   2  0.222222
1       2   3  1.000000
2       1   2  0.222222
3       1   2  0.222222
4       1   3  0.333333
5       3   1  1.000000

if you need it as a new persistent column you can do it as @jezrael proposed in the comment:

In [81]: df['ID_new'] = df.ID/df.groupby('Source').ID.transform('sum')

In [82]: df
Out[82]:
   Source  ID    ID_new
0       1   2  0.222222
1       2   3  1.000000
2       1   2  0.222222
3       1   2  0.222222
4       1   3  0.333333
5       3   1  1.000000