A选项中其实是有should 的,只是被省略了,完整的是who do you suggest (should) be sent to work there?
B 选项完整的句子是这样的who do you suggest that should be sent to work there?,仔细分析一下这个句子,you是主语,suggest 是谓语,who是宾语,而后面的that should be sent to work there是定语,说明who,而事实上,who就有somebody that的意思了,后面又有一个that就重复了.
所以不是should的问题,是that的问题