That可以引导疑问句吗?

这里涉及到两个问题：
第一个是单词inform的用法.在这个句子中用的是inform sb 加that引导的宾语从句.
第二个问题就是宾语从句.宾语从句中,从句可以是陈述句,由that引导.
eg:I know that she is your sister.(that 可省略）
我知道她是你姐姐.
也可以是一般疑问句,由if或whether引导.从句要用陈述句的语序.
eg:I don't know whether she is your sister.
我不知道她是不是你姐姐.
也可以是特殊疑问句,由特殊疑问词如what,when,how等引导,但是要注意,从句要是陈述句的语序.
eg:I am not sure when he will come.
eg:He doesn't know what her name is .
cf:He doesn't know what is her name.(这句话就是错误的.）
所以综上所述,你的问题中的那句话正确的说法应该是：
Would you like to inform me whether you have received my last email?