I can only legally carry in my vehicle in IL if I have a permit from my home state, correct?

That's my understanding...

(e) Nothing in this Act shall prohibit a non-resident from transporting a concealed firearm within his or her vehicle in Illinois, if the concealed firearm remains within his or her vehicle and the non-resident:
(1) is not prohibited from owning or possessing a firearm under federal law;
(2) is eligible to carry a firearm in public under the laws of his or her state or territory of residence, as evidenced by the possession of a concealed carry license or permit issued by his or her state of residence, if applicable; and
(3) is not in possession of a license under this Act.

If the non-resident leaves his or her vehicle unattended, he or she shall store the firearm within a locked vehicle or locked container within the vehicle in accordance with subsection (b) of Section 65 of this Act.
(Source: P.A. 98-63, eff. 7-9-13; 98-600, eff. 12-6-13; 99-78, eff. 7-20-15.)

(h) If an officer of a law enforcement agency initiates an investigative stop, including but not limited to a traffic stop, of a licensee or a non-resident carrying a concealed firearm under subsection (e) of Section 40 of this Act, upon the request of the officer the licensee or non-resident shall disclose to the officer that he or she is in possession of a concealed firearm under this Act, or present the license upon the request of the officer if he or she is a licensee or present upon the request of the officer evidence under paragraph (2) of subsection (e) of Section 40 of this Act that he or she is a non-resident qualified to carry under that subsection. The disclosure requirement under this subsection (h) is satisfied if the licensee presents his or her license to the officer or the non-resident presents to the officer evidence under paragraph (2) of subsection (e) of Section 40 of this Act that he or she is qualified to carry under that subsection. Upon the request of the officer, the licensee or non-resident shall also identify the location of the concealed firearm and permit the officer to safely secure the firearm for the duration of the investigative stop. During a traffic stop, any passenger within the vehicle who is a licensee or a non-resident carrying under subsection (e) of Section 40 of this Act must comply with the requirements of this subsection (h).

(h-1) If a licensee carrying a firearm or a non-resident carrying a firearm in a vehicle under subsection (e) of Section 40 of this Act is contacted by a law enforcement officer or emergency services personnel, the law enforcement officer or emergency services personnel may secure the firearm or direct that it be secured during the duration of the contact if the law enforcement officer or emergency services personnel determines that it is necessary for the safety of any person present, including the law enforcement officer or emergency services personnel. The licensee or nonresident shall submit to the order to secure the firearm. When the law enforcement officer or emergency services personnel have determined that the licensee or non-resident is not a threat to the safety of any person present, including the law enforcement officer or emergency services personnel, and if the licensee or non-resident is physically and mentally capable of possessing the firearm, the law enforcement officer or emergency services personnel shall return the firearm to the licensee or non-resident before releasing him or her from the scene and breaking contact. If the licensee or non-resident is transported for treatment to another location, the firearm shall be turned over to any peace officer. The peace officer shall provide a receipt which includes the make, model, caliber, and serial number of the firearm....

https://www.ar15.com/forums/Hometown/Traveling-with-a-Pistol--/23-625388/#i6893118

You can vehicle carry if you have carry in your home state. If you have some other permit that allows you to carry in your home state, it's included. There was debate time on that provision of the bill so there is legislative history, that the intent was that Vermont residents would be allowed carry without a permit. Missouri has Constitutional carry now as well as Vermont, so if it came down to it, you should be ok without a permit. Personally I would want a permit or license from some state just to be sure.