I provide a service that people can do without when they’re concerned about saving some dollars.
首先,我不得不说,你真细心!这种探求精神正是学习所需!
其次,理论上讲,【在口语和非正式语体中,关系代词在从句中作宾语时经常省略】.如：
The story (that/which)he told us is instructive.
按照此语法补充,你所给的句子【在口语和非正式语体中时】,that可以省略.
但是,【如果省略关系代词容易导致误解或费解时】,最好不省略.因为,语言的主要功能是交际,准确的传达信息才是终极目的.
最后,需补充的是,所给句子里,如果将介词without提前,那关系代词需用which,且绝对不可省略.即：
I provide a service without which people can do when they’re concerned about saving some dollars.