While the convenience of the magic shops is tempting, you decide that it would be more appropriate to acquire your ritual reagents the traditional way, by spending your time and effort to hunt down everything you need - whether metaphorically or literally. Such an action amounts to a sacrifice not all that different from the one you're planning to make to Din, which will help to strengthen and secure the familiar bond - and surely, Briar is worthy of that little extra bit of consideration.
Following the Hyrulean tradition, you require materials that represent the mystical elements which govern the land: Earth, Wind, Water, Fire, Life, Spirit, Light, Shadow, and Time.
Earth's customary offering is soil. You can get a few handfuls from the arena grounds, although a park that isn't quite so heavily shaped by human hands would offer better material, and some shrine garden would be better still. Rare earths and precious minerals are probably outside your reach, unless you're willing to resort to theft or robbery.
Wind's customary offering is... well, air. Holding the familiar ritual in some open place elevated enough to catch a light breeze would do the job, although waiting until you're in Hawaii - and away from Tokyo smog - to conduct the ceremony would likely be best. If you're willing to be more proactive, collecting bird feathers, the wings of rare insects, or leaves that have been blown off their parent tree would help.
Water's customary offering is, as one would expect, the liquid itself. You could just use tapwater or bottled water, but it'd be better to reduce the involvement and influence of human artifice as much as possible. Rainwater collected in a bowl, water scooped from a river or lake with unworked banks, seawater, or a chunk of natural ice are what you really want, especially if the source is blessed. Scales from a particularly large fish - think "great white shark" large - might also do the job.
Fire's customary offering is not so much about flame as it is the fuel. If you wanted to, you could conjure a fire and imbue it with enough of your own mana to keep it burning for the length of the ritual and have done with it, but you'd be better served by getting a brazier of coals or incense, or a branch from a particularly strong tree. Body parts from some creature with a strong association to fire, such as a dragon or phoenix, would not be remiss, but you're not going to hold your breath waiting for something like that to just drop in your lap.
Life's customary offerings are things like leaves, fruits, hair, shed scales, baby teeth - basically any part of a living being that can be lost and replaced without true harm, as long as the donor is still alive. In this case, the most powerful reagents you could use are a couple drops of your own blood and a couple of Briar's, although there might be side-effects from that. Materials harvested from some sacred animal would be almost as potent, without the potential drawbacks.
Spirit's customary offerings involve the afterlife. Swiping some soil from a fresh grave would do, as long as you didn't try to use the same soil as your Earth reagent, but some relic of a dead human body, such as bone or ash, would work better. Part of a tomb or grave-marker would also suffice, if you're willing to commit that level of vandalism; more benignly, you could entreat a priest with genuine spiritual power to whip up one of those fancy paper wards. Optimally, you want to hunt down an unquiet spirit, put it to rest, and collect whatever ectoplasmic residue it leaves behind. It's kind of a shame the Kurosakis have left; Ichigo would have been perfect bait for luring out a critter like that.
Light's customary offering is some source of illumination. Conducting the ritual at any point between dawn and noon will cover that, with the optimum results coming if you complete the ceremony precisely at noon. Calculating "true noon" is a pain, since it's not exactly the same as what Standard Time would have you believe, but the benefits would be worthwhile. Failing that, you'll need something that generates light with relatively little heat, and ideally, without using flame. A smoking torch or a flashlight would work, but a light spell would probably be better, and some bioluminescent fungus or animal or a naturally-radiant material would be best.
Shadow's customary offerings are... odd. Some practitioners handle this requirement by summoning their familiars in the twilight before dawn or at dusk or by moonlight, or in areas of natural shadow, such as the mouths of caves or under the eaves of a forest that does not quite block out the sky - any place, any time that true shadows exist in quantity. Others make use of iconography and association, giving up small personal items or creating voodoo-like representations of themselves. Summoning shadow is another option, if not the best one.
Time's customary offering is itself. The longer you spend working on gathering the other reagents, and the more time-consuming you choose to make the rituals of summoning and binding your familiar, the greater the result. This is assuming that you actually spend that time working towards your goal, and not lazing about or making a half-assed effort. For added impact, you can make a sacrifice of some item associated with Time, whether a timepiece or an antique - there were rumors that some sorcerers would go so far as sacrificing some of their own lifespan.
With all of that in mind, there are three questions that need answering.