He lives in America and has done the marriage contract with her; can she travel to join him?
If you mean that the marriage contract has been done for you and there remains nothing but consummation of the marriage, and that he wants to consummate the marriage with you there, then there is nothing wrong with that; rather it is something good and we encourage you and advise you to go ahead, because it is better for you to be with your husband and it is more chaste for you and for him, and will keep both of you further away from fitnah.
Your family and his should announce news of the consummation of the marriage everywhere and publicise it in the place where you are. It is not essential for the consummation to actually take place in the country where you are; rather that may be easier for you and it may be more affordable to get married this way in the beginning, then after that you can prepare your home gradually.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
It is better for a woman to travel overseas with her husband than to stay in her country; that is better for her and for him too. I do not see anything wrong with it.
End quote from Liqa’ al-Baab al-Maftooh, 81/17
See also the answer to question no. 3477
In that case he has to come and travel with you, or one of your mahrams should travel with you to take you to him, because a woman can only travel with a mahram. See the answer to questions no. 316 and 34380
If you mean that you will travel to him without that which is customarily called consummation, meaning that you would be travelling to him thinking that you are married in the sense that a marriage contract has been done but the marriage has not been consummated, in spite of all that you mention, and that the actual consummation of the marriage will come after he finishes his studies and gets a house and a job, when he can afford the expenses of marriage and the consequences and costs thereof, this is a kind of messing about and fooling oneself, and that is something that is not acceptable at all.
This idea of yours seems to be based on the way you phrased your question, “I got engaged on the basis of a contract” i.e., a marriage contract, as some people call this period an engagement period, even if the shar‘i marriage contract has been done.
But if that does not mean a marriage contract, then it is not permissible for you to travel to him and none of the aims of marriage result from that. Rather you should look at what your words really mean.
It says in Fataawa al-Lajnah ad-Daa’imah (18/69): Mere engagement between a man and woman does not mean that it is a marriage contract. Both the man and the woman may change their minds if they see fit during that period, whether the other party agrees or not.
See also the answer to question no. 126914
And Allah knows best.