It seems you're overstepping many workplace boundaries here.
Another more senior co-worker is working on another project, but has given me "tips" for my project.
Okay, so this person is not actually working for your project. He just offered some advice. I cannot stress this enough, and we will come back to this several times. He does not work on your project.
but that I will need him to help me on the final steps. He can either show me how to do the final steps or if that takes too long, he can do it then show me another time.
I hope you didn't phrase it like that. He's not obliged to help you, nor do you have the standing to order him to do anything (I surmise from your question).
"He can" is much too imperative. Ask if someone can and wants to help (both are relevant considerations). Don't just command them to help you.
Well today I found out that my co-worker took this entire week off and won't be back until Monday.
Were you expecting the company to ask for your permission before they approve this person's leave requests?
It also wasn't on our vacation calendar.
Are you in charge of planning the work, managing the staff, or handling payroll? No? Then there's no need for you to second-guess that someone got their leave approved, regardless whether it's short notice or not.
Remember, you don't even get to demand this person's help even if they were in the office.
I have a day off on Monday (vacation I requested weeks ago) so that means we probably won't communicate until next Tuesday. That only leaves us 2-3 days to get this complete, which is likely not enough time.
There is no "us". At least not until he has actually agreed to help you any further, and even then the project doesn't automatically become his personal responsibility. It's your project (and possibly that of others, but not him).
You have all week to work on it. Whether someone who doesn't even work on the project is present or not should not be a blocking obstacle here. If it is, that suggest that you are not assigned to a project that matches your expertise.
This project is vital to the health of the business.
That is not your call to make. That is your manager's call (or further up the chain, depending on your manager's position).
It seems like you are trying to call all the shots here. Deciding the importance of the project, deciding that your co-worker is somehow now roped into working on the project with you, and effectively proclaiming that no work can be done until they are present.
You need to take several steps back and remember your role in the organization. You have a job (doing your project). Your manager manages your workload. It seems like you are running into obstacles and blocking issues that are hampering your ability to do your job. Therefore, you should talk to your manager about the issues you're dealing with.
Do not even hint at the presumption that this coworker "must" help you, or pointing out the burden their absence puts on you. You have no standing to expect their presence or input, based on what you've told us.
At best, you can point out to your manager that this person might be able to help you if and when they might be available.
It is then up to your manager to decide whether waiting for this person's availability is the right course of action, or if other solutions present themselves. This is not your call to make.