5/24 applies to the amount of credit cards in a 24 month span not inquiries. It is very possible to have more or less inquiries than credit cards opened. You can be denied for credit, not accept a tradeline, or receive an approval with a soft pull.
Here is more info regarding Chase's 5/24 rule:
See if you can email yourself a coupon for your new bank account bonus. They offer $200 in branch but this coupon gives you $300. I recently used it myself when they offered me $200 in branch. You will most likely need to contact them to see if they can change this.
@MaizeandBlue @Janus @Aim_High @K-in-Boston @Remedios
So I have some news ...
Lots to respond to about this posting.
(1) Congrats on getting your banking set up and qualifying for the new customer bonus! This will go a long way with helping you to get approved later for a credit card. Banks usually like to extend credit to customers who have other banking products with them and especially if you demonstrate responsibility.
(2) Vantage Score credit scores generally mirror true FICO scores but they can vary widely based on a consumer's credit file. I believe I've heard some say that the more derogatory info you have, the less reliable they become to true FICO. I don't have derog info and find my Vantage Scores are pretty accurate to predicting my true FICO Bankcard Score 8's. Vantage Scores are designed to be FREE scores to give consumers a rough idea of their credit score but they are not used by any lenders to be the best of my knowledge in approving credit applications. So Chase provides that TU Vantage Score 8 but it has nothing (directly) to do with your odds of acceptance for a Chase Credit card.
(3) Hard inquires on one out of the three credit bureau's report rarely equals new cards. This is because most lenders pull only one or two of the major credit bureaus when they process an application. (Capital One is known for being an exception for their "triple pull" of all three bureaus, which affects all your FICO scores.) There are a few ways you can keep up with new cards or new inquiries. You can manually track it as-you-go in a notebook or in a computer file. You can pull your credit reports. (Free from each bureau once per year at www.annualcreditreport.com - make sure you use ONLY this website as some others are NOT free.) Another easy way to track your accounts is to set up a credit monitoring account at one of the free sites such as Credit Karma, Credit Sesame, Wallet Hub, etc. They have pages that show summaries of all your accounts including the opening date reported.
For example, in my case, I currently have 1 HP on TU, 0 HP on EQ, and 5 HP on EX. But I have six new cards opened in past two years, so none of it matches. One of those is an INK business card, so one of the EX HPs is for that card even though it doesn't show up as a 'new card' on my personal report. Four of the other pulls were single-bureau EX pulls by Chase for new cards. The single-bureau pull by Goldman Sachs was on TU for my Apple card. No one has pulled EQ in past two years. But all my FIVE new personal cards show up on all three reports.
Also, keep in mind that Inquiries are not just about credit cards. You can have HPs for loans (mortgage, auto, personal, etc.) and sometimes even from utilities such as cellphone providers who are extending you 'credit' to put a phone on extended payments.
(4) Chase always pulls your EX FICO Bankcard Score 8 and oftentimes, will also pull an EQ FICO Bankcard Score 8. Another reason why that TU Vantage Score is not applicable to getting approved for a Chase card. As I mentioned, I was established customer and they only pulled EX each time I applied for new credit. The double pull with them is more common, especially with more average scores, thinner files, or new credit customers.
*(See thread, "Which Banks Pull Which Report For Apps" list @ https://ficoforums.myfico.com/t5/General-Credit-Topics/The-quot-Which-Banks-Pull-Which-Report-For-Ap...)
(5) As pointed out by myself and others upthread, what counts towards 5/24 is new cards, not new inquiries. Only approvals, not applications.
(6) Late payments. 16? Ouch. That stings, and will for quite some time. Payment history is the dominant element of FICO scoring (35%) so that is really holding you back. Fortunately, they do age off. Fortunately, they do become less of an impact over time. Unfortunately, it takes a full SEVEN YEARS for them to age off by themselves. In some cases, you can get them removed early. I've had success sometimes with writing "Goodwill Letters" and pleading my case. Do a web search or My FICO search for "goodwill letters" to read more about them.
(7) You said, "I might still have a chance to apply for a credit card may be in 6-8 months." My advice? No, don't do it. Work on removing those lates or letting them age into the background. Set up automatic payments and make sure you don't ever make a late pay again. You have several new accounts already. Give them some love, garden and grow them for at least a year or two. Continue to build that banking relationship with Chase. Get that utilization down further. While 9% isn't bad, lower is better. And it will also help those inquries age into the background also.
(8) While I have a lot of Chase cards, I can't comment on the "Green Checkmarks" since I didn't use the preapprovals to apply.