Vedvati herself is an incarnation of Lakhsmi. Vedvati, Sita, Tulsi, Rukmani and Padmavati are the Famous incarnations of Devi Lakshmi.
Well so Devi lakshmi ever first came into this world as Vedvati. It is said that, there is a Kng named Rathadhwaja who gave up the gave up the worship of Lakshmi due to indifference. Consequently he lost the grace of Lakshmi. The enemies invaded with an army. Rathadhwaja was defeated in war and lost the kingdom. Rathashwaja had two sons Kushadhwaja and Dharmadhwaja. They felt sorry for their father's mistake. They observed rigorous penance, meditating for Lakshmi. When she appeared they prayed, "Bless us so that we get back our kingdom. Grant us a favor by being born as our daughter." Lakshmi granted the boons. They bravely fought, won the kingdom again and were living happily.
Goddess Lakshmi, in fulfillment of the boon granted by her, entered the womb of Malavati, the wife of Kushadhwaja in subtle form and was born as a daughter. It is said that even at the time of birth she was reciting Vedic verses clearly. Therefore, the parents named her 'Vedavati' Vedavati grew up to be a beautiful, virtuous girl. And started doing penance for getting Vishnu as her Husband since she is the incarnation of lakshmi.
Raavan once when wondering in Himalayas saw that beautiful Maiden performing vedic mantras and got attracted towards her. He asked her whereabouts and came to know that she is doing penance for obtaining Vishnu as her Husband. Raavan insulted Vishnu and asked her to marry him. Vedvati asked her to get lost since she cant hear a word against Vishnu. Raavan then seized hold of Vedvati by her hair with his hand. Outraged with anger, she cut of her hain by her hand and said, Insulted by U I have no more desire to live. I will take birth as the daughter of another righteous person and will myself be the cause for your death." Immediately she jumped into a burning fire in the hermitage and gave up her life. Ravana returned to Lanka grief stricken.
Vedvati later again born as Sita (which is the another incarnation of Devi lakshmi). Sita is discovered in a furrow in a ploughed field, and for that reason is regarded as a daughter of Bhumi Devi (the goddess earth). She was discovered, adopted and brought up by Janaka, king of Mithila, and his wife Sunayana. Due to all her merits which she acquired in previous birth she married to Rama(an incarnation of lord Vishnu). And later when Raavan abducts Sita in the absence of Ram, she therefore became the reason of Ravan's destruction and End and therefore has completed her vow and Taken her Revenge.
So Lakshmi Vedvati and Sita are not Different persons. Vedvati and Sita, Both r the incarnations of Devi Lakshmi who came in this world for finishing their Desired Tasks.