lack determination; cannot make decision firmly
If a person lacks determination/resolution; cannot make decision firmly:
- what words can I use to describe this?
- and how about its antonyms?

synonyms: irresolute, wishy-washy, vacillating?

antonyms: decisive, resolute, resolved, determined?

I don't think I have heard 'vacillating' in conversation but, apart from that, take your pick of the many suggestions given.

For real folk conversation on this topic, people mostly use phrases rather than individual words. He can't make up his mind (or he can never make up his mind), she doesn't know whether she's coming or going, he doesn't know how to make a decision about anything. If a one-word accusation is then added to sum up the argument, "wishy-washy" sounds like a real household word and can be used by people of any age; calling someone a "ditherer" sounds businesslike and old-fashioned and would seem hilarious coming from a child; "irresolute" sounds very academic and strongly critical (when spoken aloud); "hesitant" sounds sympathetic; and "ambivalent" sounds educated and rather detached. "Vacillating" most often refers to something that is happening at the moment; it can be used only with great difficulty to name a character flaw. "you're vacillating!" is how you most often hear it: this is said when someone is spurring you to make up your mind. As far as I know, there is no word like "vacillatory" to describe an ongoing state of vacillation. A person can, however, be called a vacillator.

Irresoluteness was one of the character flaws most dreaded in the Victorian era, and even now, to criticize someone for this vice can make one seem rather fusty and old-fashioned. In romantic matters, though, one can certainly call an ambivalent boyfriend or girlfriend "wishy-washy" without fear of seeming out of date.

For common antonym phrases, there are many possibilities, such as, "she really knows what she wants," "he has an iron will," "the woman really has willpower," "she is a forceful personality," and, more casually, "he really knows how to go for it." Single words of choice would be "determined" and "resolute." If a person is extreme in this tendency, he or she can be called intransigent, rigid, or, if you really want to get fancy, adamantine.