Welcome to the forum. I love veggies, and they constitute most of my diet. I did induction almost entirely vegetarian, and it's definitely possible and enjoyable to do Atkins as a vegetarian. I ate a bit more cheese, especially cottage (curd) cheese on induction, but otherwise I stuck to the induction without difficulty.
Coconut oil and olive oil are pretty good. You'd be hard pushed to get better.
The reason reduced fat isn't allowed is because that type of product tends to substitute the fat with carbs. You can make mayo very easily with just an egg, lemon juice, mustard and oil. Gordon Ramsey has a demonstration on Youtube.. However, if it's just a teaspoonsful, just check out the carbs, and you might find it's OK for one time use.
You can eat a load of vegetables on Atkins, and I'd suggest that you allow a little extra cheese - up to 6oz daily instead of 4oz - plus seeds and sprouted legumes etc. for your protein. Maybe you could find a way to incorporate eggs in a disguised form, because they are a tremendous food, and a great source of lecithin and great fats for your heart.. Really - it would be worth trying to find a way you enjoy them. Try making a chinese veggie stir-fry using the usual chinese flavours, such as star anise, ginger, Chinese vinegar and soy sauce, then at the end, break an egg in and stir until it is cooked. It makes the stir-fry quite creamy, and being combined with the veggies, spices, soy sauce, etc., you don't really taste the egg.
Here's my shopping list.. I will take out my meats, and add in other ideas instead. I won't add tofu as some people don't eat that.
VEGETABLES
I get an organic vegetable delivery, strong on greens, from a local farm co-operative. The contents vary according to what's in season. A recent delivery included:
kale
3 lettuce
Spinach
Spring cabbage
red cabbage
Pak choi Chinese cabbage
Mixed green salad, including watercress, mustard greens, etc,
mushrooms
aubergine (eggplant)
purple broccoli
cucumber
radishes
cauliflower
onions
leeks
celery
We'll add:
red and yellow bell pepper
garlic
ginger
hot chilis
tomatoes
lemon or lime
fresh herbs out of the garden
rhubarb from the garden
In summer, I'll have organic wild strawbs, rasps etc., fresh from the garden or the farm.
PROTEINS:
Free-range organic eggs
Fish/shellfish varies, but my top choices would be fresh sardines, smoked salmon steaks, king prawns, squid and any other oily fish
Here are some more ideas for proteins:
lentils & chickpeas etc. for sprouting; sunflower and other seeds; whey protein isolate, which you can add to your food, or make shakes with, to get extra protein
If I was you, I'd add nuts in straight after induction, but keep the quantities low. Yoghurt made from coconut milk
PRESERVED FISH (vacuum packed)
oak-smoked salmon: nice wrapped around soft cheese for a treat
oak-smoked trout: heavenly to eat straight from the pack.
smoked mackerel: we mix it with curd cheese for pate
DAIRY
Butter
Cheese (blue cheese, extra mature cheddar and parmesan).
Cottage cheese
Cream cheese - 1 tsp of luxury as an occasional snack, plus good mixed with mayo for salads
Goats cheese for cooking, and occasionally as a snack
TINS:
coconut milk
tomatoes
tuna steak
sardines
mackerel
OILS: good quality fresh olive oil, organic quality coconut oil, sesame oil, and a cheaper oil for occasional deep-frying (lard, ghee or coconut oil seem to get the best health thumb's up for frying )
CONDIMENTS, etc.:
soy sauce
Worcestershire sauce
tomato puree
Chinese vinegar or balsamic
Mustard (either dry spice or made in a jar)
Varied dried spices/herbs, always including coriander, cumin, chili, ginger, sage, mint, rosemary & either oregano or marjoram, plus a curry mix and herb blend for speed.
green tea for my breakfast
Stevia, used to sweeten rhubarb
Mayonnaise (we buy some for convenience, plus make it ourselves when there's time)
OCCASIONALLY FOR ENTERTAINING OR SPECIAL OCCASIONS:
cream (this seems to go for my blood-sugar, but we cook with it occasionally)
brandy or other cooking alcohol
greek or plain yoghurt
berries
sparkling mineral water (I limit this because of water retention)
occasional fruit for cooking with
THINGS I'D LIKE TO ADD:
Dried seaweed: I already cook with it, and everyone likes it. I am trying to find a good North Scottish source at present, as it'll be more healthy.
I can eat a load of vegetables every day and keep my carbs at induction levels by emphasizing leaves, buying fresh, buying clean (no additives apart from salt, spices, etc.), and avoiding coffee (1g carb per cup), cream (more carbs), nuts and too much cheese. I am always stuffed and my calories stay low - I often up the calories by pouring olive oil over the food.
I am eating this way for my health, but I have never enjoyed my food more, and we have never prepared delicious food faster.
Here's a lovely curry recipe - it's normally made with prawns, but you could use vegetables instead, such as cauliflower, broccoli and sugar-snap peas
http://forum.lowcarber.org/showthread.php?t=433350
I make a fabby butter curry, all vegetarian, by starting with a tin of tomatoes, onions and a variety of spices.. then adding a variety of veggies, water and nuts (optional), then before serving stir through yoghurt (you can make it from coconut milk, or use a real Greek yoghurt which is lower in carbs). If you want an approximate recipe for the amounts, let me know in my journal, and I'll write it down for you.
Good luck. The Atkins diet is great, and I have the highest regard for that wonderful man, the late great Dr. Atkins.