If there is a deal (i.e. a Withdrawal Agreement and 21-month transition period) then there will be little impact - until the end of 2010. Until then, basically little changes.
If there is No Deal, which is what the OP asked about, then things look very different. This is not scaremongering - just looking at the legal position, the logistics and explaining the consequences.
Using food as an example;
About 50% of the food in your supermarket comes from the EU; either because it is grown or made there or imported via there. 10,000 lorries and containers arrive in the UK every day carrying foodstuffs. If this traffic is curtailed, or significantly reduced, the knock-on effect is less food in the shops. If all lorries have to be checked at the arrival port, the capacity of the ports and ferries is reduced significantly. If only 3,500 lorries a day can clear the ports, then 2/3 of the imported food immediately disappears from the supermarkets. That means about a third of supermarket shelves will be empty.
50% of the food consumed is UK-produced. But it often uses ingredients from the EU. A frozen pizza is topped with Irish cheese, Danish ham and salami, Italian tomato, Dutch mushrooms etc. No ingredients means no pizzas, and other products, so another supermarket shelf is bare.
UK food is picked, processed and delivered by a workforce, of which around 50% are from the EU. If these people go home, food production is hit - crops are not picked, meat is not processed. EU commercial driving licenses become invalid in the UK, so food deliveries are reduced. Meat production is overseen by vets, who ensure that welfare and hygiene standards are maintained. 90% of these vets are from the EU. Their qualifications may no longer be valid in the UK after Brexit. No vets means no meat production, and more empty shelves.
So supply-chain problems mean that somewhere between 30% and 50% of the shelves in the supermarket will be bare - the food is simply not available to fill them.
Under normal circumstances (i.e. deliveries and supplies are normal and customers only buy what they need each week) there is about 6 days worth of goods available in the supermarkets at any given time.
In the example above, up to half of this has already become unavailable, so now the shops only hold 3 days worth. This in itself is not a problem unless people begin to panic. Instead of buying what they usually need, customers look at the empty shelves and decide to double-up on purchases, meaning that the 3 days stock is gone in a day and a half or even quicker - the less food there is on the shelves, the greater the urgency to buy a bit extra. Clearly this is only an option for those who can afford it.
The government has already stated that plans are in place to stockpile food. This has its limitations, as perishable food cannot easily be stored but there is insufficient warehouse capacity to store food in. 6 weeks worth of food would require a warehouse 6 times the size of your local supermarket - for every supermarket and shop in the country.
Stockpiling food is also only a temporary solution. Stockpiling 6 weeks worth of food just means that the country hits crisis-point in mid-May instead of at the beginning of April.
Will this actually happen? Nobody knows, but it is a distinct possibility, and the government needs to be clear about how it will mitigate the problems that this would cause.
As has happened when there have been food shortages due to the weather (e.g. snow last winter) or fuel strikes, the shops could impose their own informal rationing system, limiting customers to what they can buy.
But if food supplies really are reduced to 50% of what is required, then a more formal rationing would have to be put in place by the government to ensure that food is equally and fairly distributed and doesn't just go to those who can afford it or those with the sharpest elbows.