No reply from Haku? Yoh. Well, "most popular" words differ a bit according to context. htpp://smart.fm claims that the words used in their Core 2000 lists are the 2000 most popular words (as used in Japanese newspapers). I'd say that's a good starting point. You can download the Core 2000 and Core 6000 items for anki, too, I believe. In terms of vegetables, I don't really know, myself. A quick google search gave me two large-ish lists. The romaji one is nice and alphabetical, at least.
vegetables yasai
bamboo shoot takenoko
bean sprouts moyashi
beef steak plant (Japanese basil) shiso
broad beans soramame
broccoli burokkori
Brussels sprout mekyabetsu
burdock root (edible) gobô
cabbage kyabetsu
carrots ninjin
cauliflower karifurawâ, hanakyabetsu
celery serori
chervil (chervis) serufêyu
chicory chikori
Chinese cabbage hakusai
Chinese chive nira
chrysanthemum leaves shungiku
corn tômorokoshi
cucumber kyûri
daikon (giant white radish) daikon
daikon - yellow pickled takuan
eggplant nasu
dried gourd strips kanpyô
flowered garlic nira
garlic ninniku
ginger (fresh) shôga
ginger flower myôga
ginger shoot shinshôga
green peas gurinpîsu
green pepper (bell pepper) pîman
green pepper (minature, hot) ôba, shishitô
green shiso leaf aoshiso
leaf lettuce rîfu retasu
leeks (long) naganegi
lettuce retasu
lilly root(bulb) yurine
lotus root hasu or renkon komatsuna
mushroom masshurûmu
mushroom (brown) shîtake
mushroom (light brown/gray) shimeji
mushroom - Jew’s ear kikurage
mushroom - white button enokidake
okra okura
onion tamanegi
parsley paseri
potatoes jagaimo
pumpkin (Jp) kabocha
radish radyisshu
rape seed blossom nanohana
soy beans (fresh) edamame
soy bean feremented nattô
scallion hosonegi
snow peas sayaendô
spinach horensô
spring onion aonegi & wakegi
squash kabocha
string beans ingen
sweet potato satsumaimo
taro satoimo
trefoil (soup celery) mitsuba
turnip kabu
watercress kureson
zucchini zûkîni
yam Japanese yamaimo
1 やさい Vegetables
2 ほうれんそう spinach
3 かぼちゃ pumpkin
4 きのこ mushroom
5 きゃべつ cabbage
6 きゅうり cucumber
7 まめ beans
8 もやし bean sprouts
9 なす eggplant
10 ねぎ green onion
11 にんじん carrot
12 ににく garlic
13 ぱせり parsley
14 ぴいまん green pepper
15 れたす lettuce
16 さつまいも sweet potato
17 せろり celery
18 たけのこ bamboo shoot
19 たまねぎ onion
20 とまと tomato
21 くだもの fruit
22 あんず apricot
23 ばなな banana
24 ぶどう grape
25 いちご strawberry
26 いちじく fig
27 かき persimmon
28 めろん melon
29 みかん Japanese orange
30 もも peach
31 なし pear
32 おれんじ orange
33 れもん lemon
34 りんご apple
35 さくらんぼ cherry
36 すいか watermelon
37 うめ plum
