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Short answer: A Wyoming LLC bank account application usually rests on six papers: the filed Articles of Organization, your EIN confirmation letter (IRS CP 575), an operating agreement, a banking resolution, your passport and proof of address. Build that kit before you contact a bank, because a Wyoming LLC bank account request tends to stall on a missing or mismatched page, not on the LLC itself, and the bank alone decides.
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If you haven't formed the company yet, I'd start with CORPBOLT, whose plans include a bank-ready operating agreement and banking resolution: see CORPBOLT's Wyoming plans.
About that link: it carries my referral code, so I may be paid if you sign up through it.

Each document in a Wyoming LLC bank account kit proves one fact about the company. Collect them in this order and check each page against the last, because reviewers read them side by side.
The CP 575 is issued once. Ask the Internal Revenue Service for a 147C EIN verification letter through the Business & Specialty Tax Line at 800-829-4933, then ask whether your bank accepts it in its place.
It can be. IRS systems accept only letters, numbers, hyphens and ampersands in a business name, so "Jones.Com LLC" may come back as "Jones Dot Com LLC". Street addresses are capped at 35 characters with USPS abbreviations. Be ready to show that both papers describe one company at one address.
Know Your Customer review for a Wyoming LLC bank account is where compliance staff check the people behind the company. Expect follow-ups about every owner and signer; answer with the documents you already prepared, never versions that differ.