Sometimes, if you so choose you can offer a Mission Gear bundle or something similar that would include a 'free weapon pick' if people do not have that option. Typically, these cost a bit of experience off the base value (like -100 xp) or something based on what you feel is best. Sometimes these bundles are used as well to give players an idea of what they can expect in the mission. I like to toss in a "Bag full of Guns" gear pick into these bundles so that when it is taken, there is a gun inside it for each party member. Other things you can toss into a bundle include things like Encrypted Headset radios, various tool kits that might be important (they make a good "hint hint" item to give the characters an idea of what they may be facing), and things like Healing Cigarettes (healing gadget) or Liquid Skin patches. Ultimately, these bundles can either be generic or specially tailored for the mission you run, but they are quite handy and can be offered to single players, or as a group pick either way.
If a party member builds a character and doesn't have any weapons/charisma picks, I would recommend to them that they take the Unarmed proficiency and just deal with it, or see if any of the other party members has a spare weapon pick they can use for that player.
There are a few characters out there that could get away with no weapon picks at all, but usually as far as I have seen it just means that they haven't been creative enough with their picks so far, or they're playing a character with terrible Charisma, and it makes sense that no one will give them a weapon because nobody likes them
As far as spending cash goes, typically they are not given enough money to afford a weapon with those funds. And that money is supposed to recycle at the end of a mission. Spending Cash is effectively how much money a player has to do various small odds and ends or bureaucracy checks, gambling, and the like. My group added a house rule that allows a player to effectively stockpile their spending cash, and allows them to carry over any unspent money to the next mission. The Big Score PDF is full of ideas for using the possessions system in different ways, and I definitely suggest you pick it up while the PDF lover sale is going on. Its one of the best add on systems Crafty has made so far.