Per the Chase website, allow 6-8 weeks; if I recall correctly, I got mine in about 4.
This one-time bonus offer is valid only for first-time Marriott Rewards Credit Card Cardmembers. Previous and existing Marriott Rewards Credit Card Cardmembers are not eligible for this bonus offer. Chase is not responsible for offer fulfillment. Please allow 6 to 8 weeks after account opening for your Free Night Stay E-Certificate to be automatically deposited into your Marriott Rewards Account. Free Night Stay E-Certificate is valid for hotels in categories 1-4. For participating category 1-4 locations, log on to MarriottRewards.comOpens Overlay or call 800-321-7396.The E-Certificate must be redeemed within 6 months from date of issue. E-Certificates may not be transferred, extended beyond the expiration date, or re-credited for points.
tplyons87 Is correct, for me it was about 4 as well. One other item to note, while the cert is good for 6 months, If you call guest services they can extend it another 6 months especially if you already have a reservation during that period in which you intend to use it. I find they are very helpful.