Phase I is just safety ( and a bit of dose range finding ) - does the drug/treatment/vaccine harm or kill people on it's own - small number of people, with varying doses - ideally starting at the low end. You will do some tests to see if you get the expected biological response as a surrogate for efficacy.
Phase II is where you start looking at if it works ( while still looking at safety- always looking at safety ), here you are also trying to typically work out what is the right dosing regime ( how much, how often ).
Phase III is large enough numbers to get population level stats for safety and efficacy, given the treatment regime you have decided on.
Also you don't just need willing people to take the vaccine to do a proper trial, you need the medical infrastructure around it.
People to administer it, keep track of the patients - monitoring them for any signs of adverse reaction - reliable people to take blood samples and handle them properly, proper note and record keeping etc.
The other thing to consider in trials is that roughly half your patients don't get anything ( more accurately a placebo ) - and in the case of vaccine trials, you need to control group to get infected!!! Otherwise you can't see any protective value.
You'd need to choose you control group to carefully reflect the test group - if you had your vaccine test group in New Zealand and your control group in the US - nobody would believe your result.
At the moment people doing trials in America have an advantage in that the virus is still infecting lots of people, so you can more easily demonstrate protection. If you are trying to trials in China - pretty hard.
I don't think lack of volunteers is the problem. If you want to do it ethically, you have to do it incrementally.