This blog post contains information for Ukrainians that intend to relocate to the USA and Americans interested in hosting Ukrainians. I have identified a candidate for some space available in my home, but I have not completed this process, so I apologize for anything that is misleading or incorrect.
Americans can initiate sponsorship of Ukrainians by submitting USCIS form I-384:
There are instructions here:
The Ukrainian must supply the American with information for the first 3.5 pages of the 13-page form.
You can create an account and submit this form electronically, which describes requirements, helps to guide you through the process, and may be faster than submitting paper:
The form must be filed in the USA by the American and must include evidence of ability to provide financial support for the Ukrainian for the duration of their stay.
I assume that the American must submit a copy of this form for each Ukrainian individual, including each child. I do not know whether the children must have passports or other documentation.
Americans and Ukrainians can use the following resources to connect by offering housing and responding to such offers:
I do not like Meta/Facebook/WhatsApp, but it is useful for communications and includes translation features. The Ukrainians with whom I interact tend to speak English but google translate and similar tools may sometimes be useful.