Can I Pray without Ghusl after Wet Dreams?
Question

I am a disabled man sitting on wheelchair. Alhamdulillah.
I had a wet dream one night before Fajr prayer. Due to the weakness of my body I could not make Ghusl nor could I change my clothes except the next morning. I did not want to miss Fajr prayer on time, so I made ablution like I normally do for prayers, and then prayed. Is that permissible? If it is not then what shall I do?
Praise be to Allah.Is Ghusl required after a wet dream?
We ask Allah to grant you well-being in this world and in the Hereafter. 
Anyone who becomes Junub (a state of major impurity) as the result of a wet dream or intercourse has to do Ghusl, because Allah says (interpretation of the meaning): 
“O you who believe! When you intend to offer As-Salah (the prayer), wash your faces and your hands (forearms) up to the elbows, rub (by passing wet hands over) your heads, and (wash) your feet up to the ankles.  If you are in a state of Janabah (i.e. after a sexual discharge), purify yourselves (bathe your whole body).” [Al-Ma’idah 5:6]
What should you do if you have a wet dream and cannot do Wudu?
If one cannot do Ghusl because there is no water available, or because one is unable to use it, then one should do Tayammum and pray, because Allah says (interpretation of the meaning): 
“But if you are ill or on a journey , or any of you comes after answering the call of nature, or you have been in contact with women (i.e. sexual intercourse), and you find no water, then perform Tayammum with clean earth and rub therewith your faces and hands.  Allah does not want to place you in difficulty, but He wants to purify you, and to complete His Favour to you that you may be thankful.” [Al-Ma’idah 5:6]
Similarly, if a person has a wet dream and fears that he may become sick if he uses cold water , and there is no way of heating it, or the water is far away from him and he cannot reach it, and there is no one to bring it to him, then he may do Tayammum and pray. 
Ibn Qudamah (may Allah have mercy on him) said in Al-Mughni (1/151): 
“If a person is sick and unable to move, and there is no one who can bring him water, then he is like one who has no water, because there is no way he can get to the water. … If he has someone who can bring him water before the time for prayer ends, then he is like one who has water available.”
The sick person who cannot wash all of his body, but is able to wash part of it, must wash whatever he can then do Tayammum for the rest, because Allah says (interpretation of the meaning): 
“So keep your duty to Allah and fear Him as much as you can.” [At-Taghabun 64:16]
Based on this, you have to repeat that prayer because you did not do Ghusl or Tayammum. 
Is semen pure?
Semen is pure according to the more correct of the two scholarly opinions, hence you do not have to wash your clothes or change them. 
And Allah knows best.