Once you are separated from service, you have an absolute right to transfer or roll over your 401(k) balance into a qualified plan at a new employer or at a private IRA that you can set up with any one of several investment firms, including Vanguard. I have no idea why that phone person told you you can't take an early withdrawal. I don't believe your former employer can legally restrict you from withdrawing your funds after you have left service with them.
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I suppose they could have a policy against partial withdrawals or rollovers, and only let you do a rollover of the full amount.
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Work plans usually offer a limited selection of funds, but the funds they offer may have lower fees and expense ratios than the funds you can get privately, so it is sometimes useful to leave your funds in the work plan even after you separate. But you can always do a rollover or transfer to a new plan, even a private plan at Vanguard.
If you want to do a partial withdrawal and leave the rest in the 401(k), you can set up an IRA anywhere else, do a rollover or transfer, then do the withdrawal from the IRA. Or, you can transfer the entire balance to a new IRA, and then withdraw what you need. Investing in a new plan means you will have to select your own investments, that may require some self-education. Although a life cycle fund is a good place to park your money until you review your other options.
There is an important difference between a rollover and a transfer. With a rollover, you withdraw the money and get a check from plan 1, and you must deposit that money in plan 2 within 60 days. But it may be subject to mandatory 20% withholding. That means, if you withdraw $100,000, the check is for $80,000, but you have to deposit $100,000 in the new plan so you have to come up with the money from somewhere else. You get the withholding back in your tax refund. If you do a direct plan-to-plan transfer, you never touch the money and there is no withholding. So set up an IRA with a bank or investment firm, and tell them you want to do a plan to plan transfer out of your Vanguard 401(k). Once the money is in the new IRA, you can do a complete or partial withdrawal under the CARES act if you qualify.
It's funny because Vanguard offers IRAs. I don't see why they wouldn't offer you the option of opening an IRA and doing the plan-to-plan transfer. But maybe you don't want to stick with them since their customer support is less than stellar.
