Hi @reelscene,
Most gateways offered by Paid Memberships Pro will immediately give users membership access. Some exceptions to this are if you are using the PMPro Approvals Add On or the PMPro Pay by Check Add On which both require manual approval by an admin. Unfortunately, as those are PMPro Plus Add Ons (a premium product), we are not able to provide support for them on the www.ads-software.com forums as per the guidelines here – https://www.ads-software.com/support/guidelines/#do-not-post-about-commercial-products. Please can you open up a ticket on https://www.paidmembershipspro.com and we will continue to assist you.
As for our PayPal integrations, by default, none of them should require approval by an admin before a user gains access to your site. One notable flavor of PayPal, though, is PayPal Standard which requires PMPro to wait until PayPal approves a payment before PMPro gives membership access. For this reason, we generally caution against using PayPal Standard as that approval process can sometimes take up to a few hours. If this is the behavior that you are referring to, then we would recommend switching to another gateway, such as Stripe or PayPal Express.