To be able to collect builder nanites as well as nanite vials you will need to equip a Nanite Collector.
Now, there are several different kinds of collectors out there. Ranging from the cheap, clunky basic collector all the way up to the snazzy ultimate collector. They all have a few things in common however. That is, the statuses they provide that enables the collection of various nanites.
Let us take a look at the basic nanite collector. It's available in the basic store (viewable by typing "list items" in any room without a shop in it). It provides the following stats. (type "info basic nanite collector" in game to see the item itself.)
Nanitecollection [100% chance at 1 power for 100 rounds.]
Nanitecollection is the status that enables you to collect nanite vials. You need a total of atleast 2 to be able to actually extract nanite vials from opponents you defeat. So clearly, the simple basic collector won't help you do this. However! If you also happen to have the perk Advanced Inoculation (info Advanced Inoculation) you will gain an additional point of the nanitecollection status, enabling you to extract vials with merely the basic collector.
But not everyone wants to have that perk. Luckily, there are several other nanite collectors available that do a better job of it.
The Nanite Collector (available in the elite shop for 40 mako) has a nanitecollection rate of 2 (info nanite collector)
Lastly, there are two variants of craftable collectors. These can be made by anyone with the correct recipe, salvage and materials. These have the added benefits of being moddable to enhance the statuses they provide and to make them smaller and less of a tax on your resources to equip. These would be the Improved Nanite Collector and the Ultimate Nanite Collector.
The Improved Nanite Collector recipe is available in the token store for 100 tokens (or 20 mako) and requires a basic nanite collector to make.
The Ultimate Nanite Collector recipe is available for 20 mako in the elite store and requires a Nanite Collector (that is to say, the 40 mako variant) to make.
Both these have the additional benifit, aside from being completely moddable, to enhance your collection of builder nanites. This would be the second status you see when using the info command on either of these. NaniteCollectRate.
There are quite a few crafters about who have the improved recipe and it's not very expensive to make, making it a pretty good start for your vial collection adventures.