Vinca is usually excellent ground cover so, hopefully, if given a helping hand, it should eventually swamp the shorter oxalis. If possible you could mulch the area with compost or manure to encourage the vinca to spread (which it does by underground runners). Pull up the oxalis where possible but don't disturb the ground too much as you don't want to damage those runners.
In my experience, this plant pops up wherever it can, even though it prefers dry and sunny situations. The problem is the seed pods, they 'fire' their seeds over a fair distance, so a few plants one year translates into an awful lot more the following year. It has creeping stems that can root as they grow, but this plant is technically an annual; the problem is the prolific amount of seed it produces which ensures an ever increasing presence.
In theory, you can just pull off the foliage and flowers, but it's inevitable you will miss some and inevitable more will appear over time, so it's hard to keep up with without digging it out. Any chemical treatments you use will also kill your Vinca. It might be better to simply pull off what you can consistently (especially flowers) and then tackle it properly in Fall by digging out what you can see, even if that means digging up some Vinca and replanting it afterwards. https://www.rhs.org.uk/advice/profile?PID=498
UPDATE: on the subject of Vinca minor, I agree with the comments that this plant is not a good groundcover and behaves like an invasive weed without proper pruning. The way to keep it under control and prevent those bare areas of soil with long, sprawling stems across the empty soil is to cut it back. Every year, when the flowers are over, pull all the stems up in the air and chop them off around 2 inches above ground level. This encourages the plant to put out new growth, meaning thicker, bushier plants, with less likelihood of invasion by other weeds such as the oxalis. This assumes you actually mean V. minor and not the other plant sometimes referred to as 'vinca', Catharansus rosea; the only connection between Vinca and Catharansus is the fact they share the common name of periwinkle and belong to the Apocynaceae family, but their growth habits are quite different.
UPDATE: Vinca minor will do alright where you are, so long as you follow the pruning instructions given above, it's only required once a year, after the main flowering is over. Otherwise, Waldesteinia ternata, Campanula portenschlagiana, Heuchera sanguinea (or any other Heuchera varieties that grow well where you are), possibly Pachysandra terminalis and Catharanthus roseus, though the latter does not like temperatures that fall below 10 deg C and requires partial shade rather than full shade.
If it must go, grip it and rip it.
It's a chore, but oxalis doesn't possess nearly the caliber of immortality that some other unwelcome plants do.
On the plus side, according to https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1445-6664.2005.00167.x there is a significant correlation between above-ground Oxalis and alleopathic activity (measured vs. lettuce, iirc). So, you can potentially just rip the larger ones.
Ime, however, Ox can be banished with a thorough ripping and keeping an eye out for seedlings.