If you emailed it, then you will have to access it , on the new phone, through email.
Now if the email on both devices is the same, that should still work, you just have to not open it on the old one after sending to yourself, and instead open the email on the new device. Then choose to download the attachment.
Now, if it was actually sent from messaging/texts (not messenger), and you are opening it via email, it should still allow the download. If, however, you sent it via email or text and then are trying to open it in messaging/texts, then it may well not work.
The simplest and most clear way to do it is to upload it from the old phone to a computer by connecting the old phone to the computer via USB cord. Save the file on the computer and then connect the new phone via USB to the computer and move or copy it over to the phone. Then you just need to go into your contacts and say you want to install the info from the file.