The short answer is “no, not across the board” but the longer answer bears some explaining. First off the main small arms ammunition used will be former Warsaw Pact calibers such as 7.62×39, 5.45×39, 7.62×25 Tokarev and so forth. The point is that the calibers in the scope of concern is already small – certainly not across all small arm calibers found in the US.
Russia has a large stockpile of Russian made ammunition and plenty of production facilities to make more. Given that the US government decided not to allow new imports of Russian ammunition some time ago and importers finding other sources, this should not be a big factor in ammunition supplies.
Ukraine’s only small arms ammunition manufacturer is the Luhansk cartridge plant. Luhansk did make ammunition for the export and Century Arms sold it under the “Red Army Standard” brand but Century sources from Romania, Bosnia & Herzegovina and Poland also.
It is possible that there will be more demand for calibers the Ukrainian military uses and both the government of Ukraine and their supports will turn to the International arms market for supplies should this become a protracted conflict. That might impact supplies and prices if demand increases but supply does not.
In the US, there already is some panic buying as people who are scared/nervous decide to buy more ammo but this is on the tail end of several years of significantly high volumes of firearm and ammunition purchases already – any surge in demand will likely be short lived. Same goes with any temporary price hikes.
In closing, I really doubt there will shortages of small arms ammunition in the United States with the only caveat being that if the conflict drags on and demand in the arms market increases, there might be some price increases but only time will tell.
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Online Ammunition Sources
Note, if you haven’t checked lately, ammunition is returning to the online vendors and prices have come down dramatically from their pandemic high. For example, a 1000 rounds of bulk packed CCI 9mm Luger was over $1,000 at one point and now it is available from $299-339 sometimes with free shipping.
Also, if you haven’t heard, PSA is building an ammunition plant as well to open later this year (2022).