Is it possible to convert a file that was vectorized in Adobe Illustrator to a Photoshop .jpg file without pixelating the image when enlarging
No, a .jpg (JPEG) file is always raster and consists of discrete pixels. When you zoom enough the pixels show up. Also note a JPEG file is not a Photoshop file, it is a standard lossy raster image format.
The short answer to that would be; no.
We need to get a few facts straight to answer your question completely:
JPEG is not a Photoshop file, it's a standard lossy raster image format. There are multiple image editors that can save as JPEG.
Vector images is the use of polygons to ensure the image can be enlarged without 'pixelating'. Formats such as JPEG, PNG etc are raster images and thus can not be scaled up without losing quality.
This means there is no way to convert a vector image to raster image that can be scaled upwards. Depending on what you need the image for, there are a few things you can do. This includes saving as SVG or saving as a very large raster image and then scaling down to reduce the loss of quality. However you probably don't want very large images on ie a website.
Save it as a resolution big enough that pixellation gets unnoticeable once you magnify the image enough. For an antialiased drawing in shades of gray that means you want to save the image 256 times bigger than the intended resolution. Says you have a SVG drawing you want at 512x512 resolution, just save it as 131072x131072 image.. not very pratical. However if your "magnifier" uses bilinear interpolation the drawing just get blurred instead of pixelated.
Alternatively you could just stop zooming the image when the 131072 pixels-wide image get a 1:1 pixel correspondence to the screen, while at same time you allow editing pixels only in the 512 image.
Also note that you could get downsample artifacts when looking the image from "too far", because of that you need several versions of the image (exactly a mipmap pyramid), or some algorithm that is able to do proper filtering on the bigger image.